Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/faulttolerantgri1030lies 
 
UIUCDCS-R-80-1030 
 
 t3 
 
 UILU-ENG 80 1729 
 
 FAULT-TOLERANT GRID BROADCASTING 
 
 by 
 
 Arthur L. Liestman 
 
 ^^ 
 
 August 1980 
 
 ***** 
 
 so 
 
 
 
 
UIUCDCS-R-80-1030 
 
 FAULT -TOLERANT GRID BROADCASTING 
 
 by 
 
 Arthur L. Liestman 
 
 August 1980 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 URBANA, IL 61801 
 
 This project is funded in part by NASA Grant NSG 1471. 
 
Page 1 
 1 Introduction * 
 
 Consider a graph G = (V,E) which represents a communications network. 
 The set V of vertices corresponds to the members of the network and the set E 
 of edges corresponds to communication lines connecting pairs of members. One 
 member of the network has a message which is to be disseminated to all the 
 other members. This is to be accomplished as quickly as possible by a series 
 of calls over the network. The series of calls is constrained by the follow- 
 ing: 
 
 1. each call requires one unit of time 
 
 2. any member may participate in at most one call per time unit 
 
 3. a member can only call an adjacent member 
 
 This process is referred to as ( local ) broadcasting [Mitchell & Hedetniemi, 
 78]. In the event that a communications link fails the message may not be 
 disseminated to all of the network members. By incorporating redundancy in the 
 calling scheme the completion of the broadcast may be guaranteed in the pres- 
 ence of up to k faults. 
 
 Several aspects of the broadcasting problem have been studied. Broad- 
 casting in general graphs has been studied in [Farley, 79], [Mitchell & Hedet- 
 niemi, 78], and [Farley, et al. , 79]. Of particular interest here is broad- 
 casting in grid or grid-like graphs which has been studied in [Farley & 
 Hedetniemi, 78], [Cockayne & Hedetniemi, 78], and [Van Scoy, 79]. Grids are 
 of particular interest due to their structural regularity and frequent use in 
 parallel multiprocessor architectures. Schemes are known for minimum time 
 broadcasting from any node of a simple grid [Farley & Hedetniemi, 78]. Such 
 schemes are also known for two types of wraparound grids [Farley & Hedet- 
 niemi, 78]. The maximum number of vertices of an m-dimensional grid graph 
 which may be informed in n time Intervals Is known [Peck, 80]. 
 
Page 2 
 This paper investigates the problem of fault-tolerant broadcasting in 
 grid or grid-like graphs. Fault-tolerant broadcasting [Liestman, 80] is sim- 
 ply broadcasting with redundancy in order that certain faults, i.e. connec- 
 tions between nodes ceasing to function, do not affect the completion of the 
 broadcast. This concept is defined more formally in Chapter 2. In Chapter 3 
 fault-tolerant broadcast schemes are given for any node in a simple grid 
 graph. In Chapter 4 fault-tolerant broadcast schemes are given for nodes in 
 wraparound grids. 
 
 2 Definitions . 
 
 2 . 1 Fault-tolerant Broadcasting . 
 
 Consider a graph G = (V,E) and a subgraph G' = (V,E') with E' = E-E" 
 where E" is a set of k edges of E. The graph G represents a communications 
 network as before. The set of edges E" represents faulty communication links 
 so that the subgraph G' represents the actual communications network. Broad- 
 casting on G with enough redundancy so that the broadcast is completed when 
 any set E" of k links ceases to function is termed _k fault-tolerant 
 broadcasting . The broadcast scheme does not incorporate fault detection. The 
 standard broadcast problem corresponds to k=0. 
 
 When describing a broadcast scheme for a graph G, a call between two 
 adjacent vertices u and v at time t is represented by labelling the edge 
 (u,v) with t. If u« is the message originator and u. is another vertex of G, 
 
 a sequence of edges OtyUj), (Uj.iij), ..., (u^.u^ labelled t p tj t ± 
 
 respectively with t.<t„<...<t Ls called a calling path from u» to u. . In 
 
Page 3 
 order for u to be guaranteed to receive the message in the presence of k 
 faults there must be at least k+1 edge disjoint calling paths from u n to u. . 
 
 The notation [xj denotes the smallest integer larger than or equal to x 
 and (xj denotes the largest integer smaller than or equal to x. 
 
 2.2 Grid Graphs . 
 
 Let P, denote a path with k>2 vertices. The complete rectangular grid 
 graph of size mxn , denoted R , is defined to be the 'product' [cf. Harary, 
 69] of P and P . For example the graph below is the grid R, ,: 
 
 ' T T T T ' 
 
 > 1< ii a ii 1 
 
 > < I I II II ' 
 
 o * i ■ i < 
 
 Each vertex in R can be identified by a pair of integer coordinates (i,j) 
 
 where Ki<n, Kj<m. In drawing R in the plane let the upper leftmost ver- 
 
 m,n 
 
 tex be numbered (1,1). Let the first index increase with downward movement 
 
 and the second index increase with movement to the right. A row of R is a 
 
 ° m,n 
 
 path consisting of n vertices having the same first coordinate; a column of 
 R is a path consisting of m vertices having the same second coordinate. 
 Thus a row has a horizontal orientation and a column has a vertical orienta- 
 tion. A corner vertex of R is one of the four vertices (1,1), (l,n), 
 m,n 
 
 (m,l), (m,n) which have degree 2. A side vertex of R is of the form (i,l) 
 or (i,n) with Ki<m (called a vertical side vertex) or is of the form (l,j) or 
 (n,j) with Kj<n (called a horizontal side vertex). Side vertices have degree 
 
 3. An interior vertex of R is of the form (i,i) with Ki<m and Kj<n. 
 m,n tJ J 
 
Page 4 
 
 Interior vertices have degree 4. 
 
 A simple wraparound grid of size mxn is the grid R with additional 
 
 1 -m m,n 
 
 edges between the vertices (l»j) and (m, j) Kj<n and between the vertices 
 (i,l) and (i,n) Ki<m. This graph is denoted S . For example the graph 
 below is the grid S. •. 
 
 4,0 
 
 s 
 
 N f 
 
 ■n {• 
 
 ■\ f~\ f*\ f\ 
 
 C 
 
 
 
 
 
 
 ) 
 
 
 
 
 
 
 
 
 r k 
 
 
 
 
 
 
 ) 
 
 
 
 
 
 
 
 
 c" 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 <* 
 
 J K 
 
 J V. 
 
 J \J \J U ) 
 
 All vertices in S are similar in the sense that there is an automorphism of 
 
 m,n r 
 
 S that wil map any vertex onto any other vertex. 
 
 An Illiac wraparound grid of size mxn is the grid R with additional 
 -** m,n 
 
 edges between vertices (i,n) and (i+1,1) with Ki<m-1, between (1,1) and 
 
 (m,n), and between (l,j) and (m, j) with Kj<n. The grid is denoted I 
 
 m,n 
 
 The 
 
 following graph is I 
 
 4,6' 
 
 
 n 
 
 
 N f 
 
 V { 
 
 \ 
 
 ^ 
 
 f 
 
 ■v 
 
 S- 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 L- 
 
 
 i ► 
 
 
 
 
 
 
 
 
 
 
 J { 
 
 J 
 
 J 
 
 u 
 
 
 J ) 
 
 Illiac grids have the same property as simple wraparound grids - all vertices 
 are similar. 
 
Page 5 
 3 Fault-tolerant Broadcast Scheme s for Grids . 
 
 Consider the following problems: 
 
 1) from an arbitrary vertex (i,1) of R determine the 
 
 m,n 
 
 minimum number of time units necessary to complete 1 
 fault-tolerant broadcasting, 
 
 2) from an arbitrary vertex (i,i) of R determine a cal- 
 
 J iJ m,n 
 
 ling scheme which completes 1 fault-tolerant broadcasting 
 in minimum time. 
 
 These problems will be solved for each of the three types of vertices of 
 
 R 
 m,n 
 
 3. 1 Fault-tolerant Broadcast From a^ Corner Vertex . 
 
 Theorem 3^J_: The minimum number of time units necessary for 1 fault-tolerant 
 
 broadcasting from a corner vertex of R for m,n>5 is m+n-1 and there is a 
 
 m,n 
 
 calling scheme which completes 1 fault-tolerant broadcasting in minimum time. 
 Proof : Without loss of generality the corner vertex is (1,1). Clearly m+n-1 
 time units are required for 1 fault-tolerant broadcasting from (1,1) since 
 labelling two calling paths (of length m+n-2) from (1,1) to (m,n) according to 
 the broadcasting constraints requires at least m+n-1 time units. 
 
 The following calling scheme which completes 1 fault-tolerant broadcast 
 
 from (1,1) in m+n-1 time units completes the proof. 
 
 vertex time talks to 
 (1,1) 1 (1,2) 
 2 (2,1) 
 
 (l,j) 
 
 j-l 
 
 (I,j-D 
 
 Kj<n-1 
 
 J 
 
 (l.J+D 
 
 
 j+l 
 
 (2,j) 
 
 
 J+5 
 
 (2,j) 
 
 vertex 
 
 time 
 
 talks to 
 
 (1,1) 
 
 i 
 
 (1-1.1) 
 
 KKm-1 
 
 i+1 
 
 (1+1,1) 
 
 
 i+3 
 
 (1.2) 
 
 (m-l,D 
 
 m-1 
 
 (m-2,1) 
 
 
 m 
 
 (m,l) 
 
 
 m+1 
 
 (nrl,2) 
 
Page 6 
 
 (l,n) 
 
 n-1 
 
 n 
 
 n+4 
 
 U,n-1) 
 
 (2,n) 
 
 (2,n) 
 
 (2,j) 
 Kj<n-1 
 
 J+2 
 J+3 
 
 J+5 
 
 (l,j) 
 (3,j) 
 
 (2,j-l) 
 (2,j+l) 
 (l,j) 
 
 (2,n) 
 
 n 
 
 n+3 
 n+4 
 
 (l,n) 
 (3,n) 
 (2, n-1) 
 (l,n) 
 
 (m,j) 
 2<j<n-l 
 
 m+j-1 
 
 m+j 
 m+j+1 
 
 (ra,j-l) 
 (m,j+l) 
 (m-l,j) 
 
 (m-l,n) 
 
 m+n-3 
 nH-n-2 
 m+n-1 
 
 (m-2,n) 
 
 (m,n) 
 
 (m-l,n-l) 
 
 (i,n-l) 
 2<i<m-2 
 
 i+n 
 i+n+1 
 
 (i,n-2) 
 (i,n) 
 
 (m,n-l) 
 
 m+n— 2 
 m+n-1 
 
 (m,n-2) 
 (m,n) 
 
 (m,n) 
 
 nH-n-2 
 m+n-1 
 
 (m-l,n) 
 (m,n-l) 
 
 (2,n-l) 
 
 n+2 
 n+3 
 n+A 
 
 (2,n-2) 
 
 (2,n) 
 
 (l,n-l) 
 
 (m,l) m 
 
 m+1 
 m+4 
 
 (i,j) i+J-1 
 
 Kj<n-1 i+j 
 
 2<Km-l i+j+1 
 i+j+2 
 
 (m-l,j) m+j-2 
 Kj<n-1 m+j-1 
 m+j 
 m+j+1 
 
 (m,2) m+1 
 
 m+2 
 m+3 
 m+4 
 
 (i,n) n+i-2 
 
 2<i<m-l n+i-1 
 
 n+i+1 
 
 (m-2,n-l) m+n-2 
 m+n-1 
 
 (m-l,n-l) m+n-2 
 m+n-1 
 
 (1,11-1) n-2 
 
 n-1 
 n+4 
 
 m-1,1) 
 
 m,2) 
 
 m,2) 
 
 1-l.J) 
 
 i+l.J) 
 l.J-D 
 i,J+D 
 
 m-2,j) 
 m-l,j-l) 
 m-l,j+l) 
 m,J) 
 
 m,l) 
 m,3) 
 m-1,2) 
 m,l) 
 
 i-l,n) 
 
 i+l,n) 
 i,n-l) 
 
 m-2,n-2) 
 m-2,n) 
 
 m-l,n-2) 
 m-l,n) 
 
 (l,n-2) 
 (l,n) 
 (2, n-1) 
 
 The scheme given above Is illustrated in the following example for m=5 and 
 
n=6: 
 
 Page 7 
 
 <L^ 
 
 DS / 
 
 - a 
 
 3 
 
 . */ 
 
 ■ 5 . 
 
 X 3* *g 59 
 
 (0 6 
 9 
 
 J 
 
 * 3 6 f 
 
 r 6 ., "» .. 8 .. 9 .. /o 
 
 ¥ 
 
 5 
 , * 
 
 6 
 T 
 
 
 5 
 
 6 
 
 9 
 
 
 10 
 
 It is easily checked that this scheme adheres to the broadcast con- 
 straints. It is also easily seen that there are two calling paths from (1,1) 
 to every other vertex. For all points (i,j) with KKrn and Kj<n-1 or j=n 
 these calling paths form a rectangle with (1,1) and (i,j) at opposite corners. 
 The calling paths for (i,n-l) with Ki<n form rectangles with (1,1) and (i,n) 
 at the corners. The calling paths for (l,j) with Kj<n form a rectangle with 
 (1,1) and (2,j) at the corners. The calling paths for (i,l) with KKrn form a 
 rectangle with (1,1) and (1,2) at the corners. Thus the calling scheme com- 
 pletes 1 fault-tolerant broadcast from (1,1) in m+n-1 time units. 
 
 It should be noted that the 1 fault-tolerant broadcast scheme above 
 requires exactly 1 time unit more than the time required for fault-tolerant 
 broadcasting of [Farley & Hedetniemi, 78]. There is an additional cost 
 incurred by the 1 fault-tolerant scheme which is a higher utilization of the 
 
Page 8 
 edges (more calls). 
 
 Corollary 3_. J_: The 1 fault-tolerant broadcast scheme above requires 2nm-2m-l 
 total calls as compared to nm-1 calls for the fault-tolerant scheme of [Far- 
 ley & Hedetniemi, 78]. 
 
 Theorem 3_*2j The minimum number of time units necessary for 1 fault-tolerant 
 broadcasting from a corner vertex of R~ for n>4 is n+1 and there is a cal- 
 ling scheme which completes 1 fault-tolerant broadcasting in minimum time. 
 Proof : Without loss of generality the corner vertex is (1,1). Clearly n+1 
 time units are required since labelling two calling paths from (1,1) to (2,n) 
 according to the broadcasting constraints requires at least n+1 time units. 
 The following calling scheme which completes 1 fault-tolerant broadcast from 
 (1,1) in n+2 time units completes the proof: 
 
 vertex 
 
 time 
 
 talks to 
 
 (l.D 
 
 1 
 2 
 
 (2.1) 
 (1.2) 
 
 Cl.j) 
 
 Kj<n 
 
 J 
 j+1 
 
 J+2 
 
 (l,j-l) 
 (2,j) 
 
 d.n) 
 
 n 
 n+1 
 
 (l.n-1) 
 (2,n) 
 
 vertex 
 
 time 
 
 talks to 
 
 (2,1) 
 
 1 
 
 2 
 
 5 
 
 (1,1) 
 (2,2) 
 (2,2) 
 
 (2,j) 
 2<j<n 
 
 J 
 j+2 
 
 (2,j-D 
 (2. j+D 
 U.J) 
 
 (2,n) 
 
 n 
 n+1 
 
 (2,n-l) 
 (l.n) 
 
 (2,2) 
 
 2 
 3 
 
 4 
 
 5 
 
 (2.1) 
 (2.3) 
 (1,2) 
 (2.1) 
 
 The scheme above is illustrated in the following example for n=6: 
 
 9~ 
 
 ■J 5 
 
Page 9 
 Corollary 3^2_: The 1 fault-tolerant broadcast scheme above requires 3n-l total 
 calls as compared to 2n-l total calls in the fault-tolerant broadcast 
 scheme. 
 
 Theorem 3_.3_: The minimum number of time units necessary for 1 fault- tolerant 
 
 broadcasting from a corner vertex of R~ for n>4 is n+3 and there is a cal- 
 
 j,n 
 
 ling scheme which completes 1 fault- tolerant broadcasting in minimum time. 
 Proof ; Without loss of generality the corner vertex is (1,1). Assume that the 
 broadcast can be done in time < n+3. Consider two calling paths to (3,n). 
 Any labelling of two such paths requires at least n+2 time units. If the two 
 paths meet at a common vertex other than the endpoints the labelling requires 
 more than n+2 time units. The following are the only possible shapes that two 
 non-intersecting paths can take from (1,1) to (3,n): 
 
 <> 
 
 
 
 — ► 
 
 o 
 
 .J' 
 
 f — ' 
 
 For each shape there are two labellings of the edges so that (3,n) is informed 
 by both paths by time n+2. It Is easily checked that in each case the label- 
 ling cannot be extended so that the message can reach (l,n) from (2,1) before 
 n+3. 
 
 The following calling scheme completes the proof: 
 
 vertex 
 (1.1) 
 
 (2,1) 
 
 time 
 1 
 2 
 
 2 
 3 
 4 
 
 talks to 
 
 (1,2) 
 
 (2,1) 
 
 (1,1) 
 (3,1) 
 (2,2) 
 
 vertex 
 
 time 
 
 talks to 
 
 (l.J) 
 
 j-l 
 
 (l.J-1) 
 
 Kj<n-1 
 
 j 
 
 (l.j+l) 
 
 
 JH 
 
 (2,j) 
 
 
 J+5 
 
 (2,j) 
 
 (2, J) 
 
 J*-l 
 
 d.j) 
 
Page 10 
 
 (3,1) 
 
 (3,2) 
 
 (l,n-l) 
 
 (2,n-l) 
 
 (3,n) 
 
 3 
 
 (2, 
 
 1) 
 
 4 
 
 (3, 
 
 2) 
 
 7 
 
 (3, 
 
 2) 
 
 4 
 
 (3, 
 
 1) 
 
 5 
 
 (3, 
 
 3) 
 
 6 
 
 (2, 
 
 2) 
 
 7 
 
 (3, 
 
 1) 
 
 n-3 
 
 (1, 
 
 n-2) 
 
 n-2 
 
 (1, 
 
 n) 
 
 n+3 
 
 (2, 
 
 n-1) 
 
 n+1 
 
 (2, 
 
 n-2) 
 
 n+2 
 
 (2, 
 
 n) 
 
 n+3 
 
 (1, 
 
 n-1) 
 
 n+1 
 
 (2, 
 
 n) 
 
 n+2 
 
 (3, 
 
 n-1) 
 
 Kj<n-1 
 
 J+2 
 
 (2 
 
 .j-l) 
 
 
 J+3 
 
 (2, 
 
 J+l) 
 
 
 *■<* 
 
 (3, 
 
 J) 
 
 
 J+5 
 
 (1 
 
 J) 
 
 (3,j) 
 
 ^2 
 
 (3, 
 
 J-D 
 
 2<j<n-l 
 
 J+3 
 
 (3, 
 
 ,j+D 
 
 
 ^"4 
 
 (2, 
 
 J) 
 
 (3, n-1) 
 
 n 
 
 (3, 
 
 n-2) 
 
 
 n+1 
 
 (3, 
 
 n) 
 
 (l,n) 
 
 n-1 
 
 (1, 
 
 n-1) 
 
 
 n 
 
 (2, 
 
 n) 
 
 
 n+3 
 
 (2, 
 
 n) 
 
 (2,n) 
 
 n 
 
 (1, 
 
 n) 
 
 
 n+1 
 
 (3, 
 
 n) 
 
 
 n+2 
 
 (2, 
 
 n-1) 
 
 
 n+3 
 
 (1, 
 
 n) 
 
 The scheme above is illustrated in the following example for n=6: 
 
 < 
 
 ft ' i 
 
 a 
 
 , ^ , 
 
 , H 
 
 5 
 
 a 
 
 r 
 
 3 
 
 • \ 
 
 8 5 
 
 , b i 
 
 <> 
 
 9 (,' 
 
 . 8 
 
 3 
 
 
 
 8 
 
 -U 
 
 > r < 
 
 Corollary 2.*3_ : The 1 fault-tolerant broadcast scheme above requires 6n-5 calls 
 a-s compared to 3n-l calls in the fault-tolerant broadcast scheme. 
 
 Theorem 3*4: The minimum number of time units necessary for 1 fault-tolerant 
 
 broadcasting from a corner vertex of R, for n>6 is n+3 and there is a cal- 
 
 **, n 
 
 ling scheme which completes 1 fault-tolerant broadcasting in minimum time. 
 
Proof : Without loss of generality the corner vertex is (1,1) 
 time units are required to label calling paths to (4,n). 
 
 Page 11 
 Clearly n+3 
 
 The following calling scheme completes the proof: 
 
 vertex 
 
 time talks 
 
 to 
 
 (1. 
 
 1) 
 
 1 ( 
 
 2 ( 
 
 :i,2) 
 :2,d 
 
 
 (2, 
 
 1) 
 
 2 ( 
 
 3 < 
 
 4 ( 
 
 :i,d 
 :3,d 
 :2,2) 
 
 
 (3, 
 
 1) 
 
 3 ( 
 
 4 < 
 
 5 ( 
 9 ( 
 
 :2,d 
 :3,2) 
 :4,d 
 
 [3,2) 
 
 
 (4, 
 
 1) 
 
 5 ( 
 
 6 I 
 9 < 
 
 :3,d 
 :4,2) 
 :4,2) 
 
 
 (3, 
 
 2) 
 
 4 1 
 
 5 1 
 
 6 ( 
 
 8 1 
 
 9 1 
 
 :3,d 
 :3,3) 
 :2,2) 
 :4,2) 
 :3,d 
 
 
 (4,2) 
 
 6 < 
 
 :4,d 
 
 
 
 
 7 1 
 
 :4,3) 
 
 
 
 
 8 1 
 
 :3,2) 
 
 
 
 
 9 I 
 
 :4,d 
 
 
 (2, 
 
 n-1) 
 
 n+1 1 
 n+2 1 
 
 '2,n- 
 :2,n) 
 
 2) 
 
 
 
 n+3 1 
 
 :i,n- 
 
 1) 
 
 (3, 
 
 ,n-l) 
 
 n+1 1 
 
 '3,n- 
 
 2) 
 
 
 
 n+2 
 
 [4, n- 
 
 1) 
 
 
 
 n+3 i 
 
 :3,n) 
 
 
 (1 
 
 >n) 
 
 n-1 
 
 n 1 
 n+3 
 
 :2,n) 
 (2,n) 
 
 1) 
 
 (4 
 
 ,n-l) 
 
 n+2 
 n+3 
 
 (3,n- 
 (4,n) 
 
 1) 
 
 vertex 
 
 U,j) 
 
 Kj<n-1 
 
 time t 
 j-1 < 
 J ( 
 j+1 < 
 J+5 < 
 
 :alks to 
 
 :i,j+D 
 
 :2,j) 
 
 :2,j) 
 
 (2,j) 
 Kj<n-1 
 
 +H < 
 j+2 < 
 j+3 ( 
 +H4 < 
 j+5 ( 
 
 [2, j-1) 
 
 [2, j+1) 
 3.J) 
 
 :i,j) 
 
 (3,j) 
 2<j<n-l 
 
 ;H-2 1 
 
 j+3 < 
 
 JH-4 < 
 n+3 ( 
 
 :3,j-D 
 :3,j+d 
 :2,j) 
 >,j) 
 
 (4,j) 
 2<j<n-2 
 
 +H4 1 
 
 j+5 < 
 n+3 ( 
 
 :a,j+d 
 :3,j) 
 
 (4,n-2) 
 
 n+2 1 
 n+3 ( 
 
 :4,n-3) 
 :,3,n-2) 
 
 (l,n-l) 
 
 n-2 1 
 n-1 1 
 n+3 1 
 
 :i,n-2) 
 
 :i,n) 
 
 :2,n-l) 
 
 (2,n) 
 
 n i 
 n+1 < 
 n+2 1 
 n+3 1 
 
 :i,n) 
 
 :3,n) 
 [2, n-1) 
 [l.n) 
 
 (3,n) 
 
 n+1 ( 
 
 n+2 I 
 n+3 1 
 
 :2,n) 
 :4,n) 
 :3,n-l) 
 
 (4,n) 
 
 n+2 1 
 n+3 1 
 
 :3,n) 
 :4,n-l) 
 
Page 12 
 
 The scheme above is illustrated in the following example for n»7: 
 
 f 
 
 B ' V 
 
 a. 
 
 3 
 
 4 
 
 5 
 
 4 
 
 
 
 3 ? <*e 51 t\o 10 tlo 
 
 , H r S 4 7 . 8 .. 1 
 
 3 
 
 *,1 
 
 * 8 9 
 
 S i. 7 8 
 
 8 
 
 5 
 l 
 
 8 
 
 ID 10 10 9 
 
 1 
 
 > 
 
 Corollary 3_.4_: The 1 fault-tolerant broadcast scheme above requires 8n-6 calls 
 as compared to An-1 calls in the fault-tolerant scheme. 
 
 Theorem 3_.jk The minimum number of time units for 1 fault-tolerant broadcast 
 
 from a corner vertex of R for the cases: 
 
 m,n 
 
 1. m=2, n=2,3 
 
 2. m=3, n=3 
 
 3. m=4, n=4,5 
 
 is m+n and there is a calling scheme which completes 1 fault-tolerant broad- 
 cast in minimum time. 
 
 Proof : The proof that m+n units are required in each case is omitted. The 
 arguments are similar to those above. 
 
The following schemes complete the proof: 
 
 Page 13 
 
 ■i-^2- 
 
 *,5 
 
 t 
 
 » > 
 
 a 
 
 3 3 
 
 ■ ■ v • 
 
 6 3 6 
 
 . -5 . 
 
 3 
 
 i 
 
 ^JL. 
 
 
 /" 
 
 D ' < 
 
 i 2 j 
 
 i ■» . 
 
 Z 
 
 3 
 
 s 
 
 8 8 
 
 b 
 
 
 3 
 
 6 
 
 7 i 
 
 5 
 
 8 , 
 
 
 5 
 
 ■ * , 
 
 6 
 ^.8 . 
 
 
 ^ / 
 
 . 2 
 
 . 3 
 
 . S 
 
 
 a 36 ? 
 
 V s 
 
 9 8 58 
 6 ? 
 
 3 
 
 
 8 £ 
 6 ( * 8 
 
 i 
 
 8 
 
 9 
 
 i 1 
 
 7 
 i « 
 
 7 
 
 , 6 . 
 
 ► 
 
 Corollary 3_.J5: The 1 fault-tolerant schemes above require the following number 
 of total calls: 
 
 1 . mn+n- 1 
 
 2. 2mn-m-l 
 
 3. 2mn-2m 
 
 4. 2mn-m-l 
 
 for m=2, n=2, 3 
 
 for m=3, n=3 
 
 for m=4, n=4 
 
 for m=4, n=5 
 
 as compared to mn-1 for the fault-tolerant scheme. 
 
 3.2 Fault-tolerant Broadcast from a Side Vertex. 
 
 The above results can be used to construct schemes for 1 fault-tolerant 
 broadcasting from a side vertex. Without loss of generality the message ori- 
 ginator is vertex (i,l) with Ki<[m/2J. The vertex (i,l) is the corner vertex 
 of two subgrids ^ n and ^.^+1 n which share the edges and vertices in row i. 
 For the remainder of this section we will refer to the former as the upper 
 
Page 14 
 
 subgrid and the latter as the lower subgrid. The two share row 1 which Is row 
 
 1 of each of the subgrids. The general approach for broadcasting from (i,l) 
 
 will be to use the corner scheme for R ... given above for the lower 
 
 m-i+l,n ° 
 
 subgrid and a modified version of the corner scheme for R. given above for 
 
 l,n 
 
 the upper subgrid. 
 
 As an illustration, consider vertex (4,1) of Rg ,. as the message origina- 
 tor. The lower subgrid is R, _. The broadcast in the lower grid is as fol- 
 
 o, :> 
 
 lows: 
 
 
 B ' . 
 
 X 
 
 3 
 
 H 
 
 
 1 
 
 p—- ■ ■ < 
 
 3 
 
 5 
 
 1 H- 
 
 8 
 
 I ? - 
 
 f S 
 
 8 , 
 
 f 
 
 3 
 
 
 5 
 , 7 
 
 8 
 
 f f 
 
 
 H 
 
 5 
 ■ 7 
 
 6 
 
 8 
 
 9 
 
 ? 
 
 10 
 
 
 5 
 
 (p 
 
 r f , 
 
 8 
 
 <? 
 
 S 
 
 10 
 
 
 k 
 
 1 1 ! 
 
 to 
 
 
 9 
 
 Id 
 
 i « 
 
 1 
 
 The upper subgrid uses the scheme given above for R, ,. with the following 
 modifications: 
 
 1. All calls except those between row 1 vertices which 
 occur at time t are delayed until time t+1. 
 
 2. All calls between (l,j) and (2,j) which occur later 
 than time n+1 (after the delay) are deleted. 
 
 The delay introduced in 1 avoids the problem of vertices in the shared row 
 calling in two directions at once. The calls deleted by 2 were needed to com- 
 plete a second path to row 1 vertices in the original scheme for the upper 
 subgrid. These calls are not needed here since the second path to each of the 
 
Page 15 
 shared row vertices Is Included In the R, ,. scheme. The resulting scheme is: 
 
 
 (> . 
 
 , 7 
 
 . e 
 
 9 
 
 
 5 
 
 9 
 
 , 6 , 
 
 h 
 
 7 
 
 8 , 
 
 8 
 
 , 1 
 
 
 1 
 
 
 
 6> 
 
 ?■ 
 
 7- 
 8 , 
 
 
 3 
 
 
 5 
 3 
 
 3 
 
 6 
 ^ 
 
 
 5 
 2. 
 
 5" 
 
 r 6 ( 
 
 8 
 
 , * 4 
 
 ? 5 
 
 9 
 
 3 
 
 4 
 
 s 
 
 & 
 
 6 
 
 
 H 
 
 7 j 
 
 6 
 
 9 
 
 
 
 s 
 
 
 8 , 
 
 <? 
 
 6 
 
 ID 
 
 
 ■ 
 
 9 
 
 
 _3 , 
 
 1 
 
 < < 
 
 
 This general approach be used for 1 fault-tolerant broadcasting from any 
 side vertex (i,l) with Ki<[m/2] of R provided that the subgrid schemes can 
 be combined in this manner without conflict. 
 
 Most of the corner schemes have the edges of row 1 labelled 1, 2, ..., 
 
 n-1. For the general approach such a labelling will be required; thus a 
 
 corner scheme with row 1 labelled 1, 2, ..., n-1 must be determined for all 
 
 grids R . Define t(m,n) to be the time required to broadcast from the 
 ° m,n 
 
 corner vertex of R given that calls are made along row 1 at times 1, 2, 
 
 m,n ° ° 
 
 .., n-1. From the above results: 
 
 m+n-1 
 
 t(m,n) = 
 
 m+n 
 
 for m,n>5 
 
 m=4, n>6 
 m>4, n=2 
 
 for m=2, 3, n=2 
 m=3, n>3 
 m=4, n=»4,5 
 
Page 16 
 
 The value is not apparent for other values of m and n since the known schemes 
 
 for these grids are not of the proper form. 
 
 Theorem 3^6_: 
 
 m+n+1 m=2, n=3 
 
 t(m,n) = m+n m=2, n>4 
 
 m>4, n=3 
 m>5, n=4 
 
 Proof ; Consider such a broadcast from the corner of R« «. In order to get two 
 paths to (1,3) by time 5 the following calls are required: 
 
 L-3- 
 
 There is no extension of this labelling which includes 2 calling paths to 
 (2,1) by time 5. The following scheme informs all vertices by time 6: 
 
 *- 
 
 / , -a 
 
 3> 
 
 Consider a grid R with n>4. Clearly if the top row is labelled with 
 
 ° 2,n ' r 
 
 1, 2, 3, • .., n-1 then a second path cannot reach (l,n) until time n+2. The 
 following scheme informs all vertices by time n+2: 
 
 «) 1 f 
 
 J 5 (. 
 
 (i 1 «! 
 
 3,fr H 
 
 n-3 
 
 f "~*\ 
 
 I °"' I 
 
 i 
 
 
 i 1 
 
 n-l n 
 
 n-hi 
 
 Consider a grid R _ with m>4. To label 2 calling paths to (m, 3) requires at 
 m, 5 
 
Page 17 
 least m+2 units. In order to extend such a labelling so that there are also 
 two calling paths to (m, 1) at least m+3 units are required. The following 
 
 scheme informs all vertices of R „ by time m+3: 
 
 m,3 ' 
 
 
 / 
 
 2 
 
 
 3 
 
 5 
 
 3 
 
 l 5 ^ , 
 
 b 
 
 H 
 
 4 l* , 
 
 * , 
 
 5 
 
 
 
 
 , »»»,*»» , , mtl 
 
 t»H-l 
 
 
 ultZ 
 
 wtt 
 
 rr,-l 
 
 OO 
 
 mo+i 
 
 mn*3 
 
 Consider a grid R , with m>5. To label two calling paths to (m, 3) requires 
 at least m+3 units. In order to extend such a labelling so that there are 
 also two calling paths to (m, 1) at least m+4 units are required. The follow- 
 
ing scheme informs all vertices of R , by time m+4: 
 
 in } "f 
 
 
 „ / 
 
 3 
 
 3 
 
 (r - 
 
 
 
 i 
 
 3 
 
 
 t 
 
 
 5 
 
 b 
 
 , ? , 
 
 1 < ■ 
 3 * 
 
 
 5 
 
 
 b . f 
 
 « , 
 
 V- S 
 
 
 6 
 
 Page 18 
 
 m-A 
 
 «»-l 
 
 ► " 1+ » l i 
 
 . "lO 
 
 r*-l 
 
 1 IW1 ' 
 
 w+i , 
 
 "H-3 . 
 
 4 
 
 1 ' 
 
 r<+3 
 
 r ni - t '' 2 - ■ 
 
 r v^t-3 
 
 
 ^-^ 
 
 
 
 *v> 
 
 vn-H 
 
 m+T- 
 
 Corollary 3^6_: The 1 fault-tolerant schemes given above with the edges of row 
 1 labelled 1, 2, ..., n-1 require the following number of total calls as com- 
 pared to mn-1 in the fault-tolerant schemes: 
 
 3n-l 
 
 for m>l, 
 
 n=2 
 
 9 
 
 m"2, 
 
 n=3 
 
 3n-l 
 
 m=2, 
 
 n>3 
 
 13 
 
 m=3, 
 
 n=3 
 
 6n-5 
 
 m=3, 
 
 n>3 
 
 5m-l 
 
 m>3, 
 
 n=3 
 
 24 
 
 nr=4, 
 
 n=4 
 
 35 
 
 m=4, 
 
 n«5 
 
 8n-6 
 
 m=4, 
 
 n>5 
 
 6m- 1 
 
 m>4, 
 
 n= 3 4 
 
 2nm-2m-l 
 
 m>4, 
 
 n>4 
 
 Theorem 3_.7j There Is a calling scheme which completes 1 fault-tolerant broad- 
 cast from a side vertex (i,l) with Ki<[m/2) of R in minimum time. 
 
 L J m,n 
 
 Proof : In any scheme for 1 fault-tolerant broadcasting from (i,l) there must 
 be two calling paths to each of the corners (m,n) and (l,n). To label 2 paths 
 
Page 19 
 to (m,n) requires m-i+n time units as the distance from (i,l) is m-i+n-1. If 
 m=2i-l then the distances to the two corners are equal and at least i+n units 
 are required to label the paths to the corners. Thus any scheme must take m- 
 i+n time units (or i+n if m=«2i-l) to complete 1 fault-tolerant broadcast on 
 
 R 
 
 m,n 
 
 A calling scheme can be constructed for any side vertex of R as 
 
 ° m,n 
 
 described above. The method for combining the corner schemes for two subgrids 
 is as follows: 
 
 1. apply the 1 fault-tolerant corner broadcast scheme for 
 
 R ... with row 1 edges labelled 1, 2, . .., n-1 to the 
 m— l+i, n 
 
 lower subgrid. 
 
 2. apply the 1 fault-tolerant corner broadcast scheme for 
 R. with row 1 edges labelled 1, 2, ..., n-1 to the upper 
 subgrid with the following modifications: 
 
 a. all calls except those between row 1 vertices which 
 occur at time t are delayed until time t+1. 
 
 b. unless 1=2 and n>3 all calls between (l,j) and (2,j) 
 which occur later than time n+1 (after the delay) are 
 deleted. 
 
 Using the 1 fault-tolerant corner broadcast schemes with row 1 labelled 1, 2, 
 ..., n-1 which attain the broadcast times t(i,n) and t(m-i+l,n) a scheme for 1 
 fault-tolerant broadcasting from (1,1) is constructed. It is easily verified 
 that there are two calling paths to each vertex and that none of the broadcast 
 rules are violated. The time required by this scheme is easily seen to be no 
 greater than max{t(m-i+l,n), t(I,n)+l}. 
 
 The time required by the constructed schemes Is actually: 
 
i+n+2 
 i+n+1 
 
 if i=2, m=3,4 n=3 
 
 if i=2, m=3, n>3 
 
 i>3, m=2i-l, n-3 
 
 i>4, m=2i-l, n-4 
 
 i=4, m=7,8, n=5 
 
 1=2, ra=4, n=2 
 
 Page 20 
 
 i+n 
 
 if 
 
 m- i+n+1 
 
 m-i+n 
 
 if 
 
 if 
 
 i=2 
 
 , m=3, 
 
 n=2 
 
 1=3 
 
 . m=5, 
 
 n=2 
 
 1=3 
 
 , m=5, 
 
 n>4 
 
 i=3 
 
 , m=6, 
 
 n>6 
 
 1=4, 
 
 m=7, 
 
 n>6 
 
 i>4 
 
 , m=2i- 
 
 -1, n=2 
 
 i>5 
 
 , m=2i- 
 
 -1, n>5 
 
 1=2, 
 
 m>5, 
 
 n=3 
 
 i>2 
 
 , m>2i 
 
 , n=4 
 
 i=3 
 
 , m=6, 
 
 n=5 
 
 1=3, 
 
 m>6, 
 
 n=3 
 
 i>4 
 
 m>2i 
 
 , n=3 
 
 1=2, 
 
 m=4, 
 
 n>4 
 
 1=2 
 
 , m=5, 
 
 n=2 
 
 1=2, 
 
 m=5, 
 
 n>6 
 
 1=2, 
 
 m>6, 
 
 n>5 
 
 1=3, 
 
 m>6, 
 
 n=2 
 
 1=3, 
 
 m>7, 
 
 n>5 
 
 1=4, 
 
 m>9, 
 
 n=5 
 
 i>4 
 
 , m>2i 
 
 > n=2 
 
 1=4, 
 
 m>8, 
 
 n>6 
 
 i>5 
 
 , m>2i 
 
 , n>5 
 
 In some cases (such as 1=2, m=3, n=2) the resulting time is less than 
 max{ t(m-i+l,n), t(i,n)+l} due to the deletion of calls by step 2b. The 
 schemes constructed for the cases i=2, m=3,4, n=3 and i=3, m=3, n>3 can all be 
 improved. The constructed schemes are: 
 
 
 1? 
 
 .3 
 t 
 
 1 ' i 
 
 31 
 
 i 
 
 z 
 i 
 
 
 . * 
 
 H 
 
 3.5 
 
 
 ^ 
 
 5 
 
 
 3> 
 
 1 
 
 a 
 
 4 
 
 1 
 
 2 
 
 J * 
 
 <o 
 
 1 1 i 
 
 3 
 
 , 5 
 
 C 
 
 3 
 
 4 
 
 JLkn 
 
 H 
 
 > 
 
 
 H**. 
 
 5 
 
 .6 
 
 3 
 
 c 
 
 f 
 
 
 
 i ' « 
 
 r ^ , 
 
 ,-3 
 
 i 
 
 7 ' 
 
 
 
 7 
 
 5 
 
 C 
 
 
 i 
 
 V 1 
 i — i 1 
 
 , 1 .< 
 
 ,s 
 
 "» , 
 
 r>+i 
 
 n + a 
 
 ■*-3 
 
 
 n+3 
 
 . — — i 
 
 i < 
 
 n+a 
 
 1 1 
 
 \rt n-f/ 
 
Page 21 
 In the first two cases the deletion of labels by 2b allows a relabelling of 
 other edges resulting in the following schemes: 
 
 V. <. 5 . ft 5 
 
 «- 
 
 JAi 
 
 3.5 
 
 *~ 
 
 jjhk_l 
 
 In the case 1=2, m=3, n>3 the following relabelling is possible: 
 
 <^- 
 
 3,fe 
 
 A^^ 
 
 ri 
 
 n+/ 
 
 n+z 
 
 r%-C 
 
 »1 
 
 n 
 
 n-3 ( 
 
 »-l 1 
 
 o-l 
 
 *+l 
 
 M+Z 
 
 o+i 
 
 *i-l 
 
 v\ 
 
 ' »+i ' 
 
 Thus i+n+1 units are used by the schemes for 1=2, m=3,4, n=2 and i+n units are 
 used for 1=2, m=3, n>3. 
 
 The schemes which take m-i+n units and the schemes which take I+n units 
 if m=2i-l are minimum time schemes. The other cases can also be shown to be 
 as fast as possible. Consider, for example, the case 1=3, m>6, n=3. The 
 lower subgrld is R, _ with k>3. The proof of Theorem 3.6 shows that to com- 
 plete 2 calling paths to both (k,l) and (k,3) requires k+3 units. Thus k+3 = 
 m-i+n+1 units are needed for the side vertex problem. Similar arguments hold 
 for the remaining cases. 
 
Page 22 
 3 . 3 Fault-tolerant Broadcasting from an Interior Vertex . 
 
 As In the previous section the corner broadcast schemes may be combined 
 
 to create a 1 fault-tolerant broadcast scheme for interior vertices of a grid. 
 
 Without loss of generality the message originator is vertex (i,j) with 
 
 Ki<[m/2] and Kj<[n/2J. The vertex (i,j) is the corner vertex of four 
 
 subgrids R. . , R, . , , , R ,., ., and R ... „ . , , . The general approach for 
 & i,j i,n-j+l' m-i+l,j' m-i+l,n-j+l ° rr 
 
 broadcasting from (i,1) will be to use the corner scheme for R ... ... 
 
 6 ,J m-i+l,n-j+l 
 
 given above and modified versions of the corner schemes for the other 
 subgrids. The corner schemes given above which have the edges of row 1 
 labelled 1, 2, ..., n-1 also have the edges of column 1 labelled 2, 3, . . . , m. 
 This will be required for the general approach. 
 
 Let t(ra,n,i,j) be the minimum time required for 1 fault-tolerant broad- 
 casting from an interior vertex (i,j) of R with Ki<[m/2] and Kj<[n/2j. 
 
 id * n 
 
 Theorem 3.8: t(m,n,i,j) > m+n-i-j+1. 
 
 Proof : The distance from (i,j) to (m,n) is m+n-i-j. To complete two calling 
 
 paths to (ra,n) requires m+n-i-j+1 time units. 
 
 Theorem 3_.^9: There is a calling scheme which completes 1 fault-tolerant broad- 
 cast from any vertex (i,j) of R with 5<i<[m/2J and 5<i<[n/2J in time 
 
 m,n LJ ->lj 
 
 max{m+n-i-J+l, n+i-j+1, m+j-i+3, i+j+3}. This is known to be minimum time for 
 
 m>2i and n>2j+2. 
 
 Proof: Consider the grid R . The four subgrids R J Jf R. ,,,, R .,, ., and 
 m,n B i,j» i,n-j+l' m-i+l,j' 
 
 R -1+1 -i+i determined by the message originator (i,j) will be referred to as 
 the upper left, upper right, lower left, and lower right subgrids respec- 
 tively. The corner broadcast schemes used on each subgrid will assume an 
 
Page 23 
 orientation in which row i of the grid becomes row 1 of the subgrid and column 
 j of the grid becomes column 1 of the subgrid. A calling scheme can be con- 
 structed as follows: 
 
 1. Apply the 1 fault- tolerant corner broadcast scheme for 
 
 R ... ... to the lower right subgrid. The call made 
 
 m-i+l,n-j+l e & 
 
 between (m-i+1,1) and (m-i+1,2) at time m-i+5 is delayed by 
 one time unit. 
 
 2. Apply the 1 fault- tolerant corner broadcast scheme for 
 
 R ... to the lower left subgrid. All calls except those 
 m-i+l,j 
 
 between column 1 vertices are delayed by three time units. 
 (The times of the column 1 calls are determined by the lower 
 right subgrid.) The second call between (m-i+1,1) and (m- 
 i+1,2) is deleted. 
 
 3. Apply the 1 fault- tolerant corner broadcast scheme for 
 R. _j.i to the upper right subgrid. All calls except those 
 in row 1 are delayed by one time unit. (The times of the 
 row 1 calls are determined by the lower right subgrid.) The 
 second call between (i,l) and (i,2) is delayed by one addi- 
 tional time unit. All calls which occur between row 1 and 
 row 2 after time n-j+2 are deleted. 
 
 4. Apply the 1 fault-tolerant corner broadcast scheme for 
 
 R. . to the upper left subgrid. All calls except those 
 i » J 
 
 between column 1 vertices and those between row 1 vertices 
 are delayed by four time units. (The times of the row 1 and 
 column 1 calls are determined by the lower left and upper 
 right subgrids respectively.) All calls which occur between 
 row 1 and row 2 after time j+5 or between column 1 and 
 
Page 24 
 
 column 2 after time i+5 are deleted. 
 
 The scheme is illustrated by the following example. The grid is R 
 and the message originator is (5,5). The schemes for each subgrid before 
 delays are: 
 
 
 <? 
 
 , 9 
 
 , 7 
 
 .4.1 , 
 
 
 8 
 
 i 9 j 
 
 , 8 . 
 
 • 
 
 9 
 
 7 , 
 
 6 . 
 
 & 
 
 7 
 
 1 
 
 
 , 7 
 
 6 
 
 V 
 
 U 
 
 i 8 ( 
 
 5 
 
 H 
 L 
 
 , 5 
 
 3 
 
 5 
 
 4 
 
 1 t 
 3 
 
 6 3 
 
 .2 . 
 
 7 
 
 z 
 
 
 <* 3 . 
 
 1 
 
 1 t 
 
 S 
 
 1 
 
 8 
 
 1 4 
 
 7 
 
 8 3 
 
 7 
 
 5 , 
 
 6 
 
 1 , 
 
 5" 
 8 , 
 
 V 
 
 7 
 
 4 
 
 3 
 
 ? 
 
 IO 
 
 6 
 
 S 
 
 7 
 
 V 
 
 8 
 
 . Id 
 
 7 
 
 f 1 I 
 
 6 
 
 , 8 
 
 , 7 , 
 
 5 
 
 5 
 
 '0 
 
 
 1 
 
 . V?.- 
 
 i 
 
 
 . */? 
 
 . 7 
 
 . 8 _ 1 
 
 . '° 
 
 
 5 9 to «? 
 
 
 V 
 
 5 
 
 6 
 
 , 7 , 
 
 7 
 
 i 8 J 
 
 , ? t 
 
 8 
 
 3 
 
 V 
 
 , 5 
 
 5 
 
 , 7 
 
 r 8 „ ? ,■ 
 
 a. 
 
 ( 
 
 3 
 &— ' 
 
 7 4 
 
 z 
 l < 
 
 8 s 
 
 f 10 i IP 
 
 
 / a. 3 4 
 
 5 
 
 
 o , 
 
 3 
 
 , -5 , 
 
 7 » 
 
 , 6 , 
 
 8 S 
 , 7 
 
 
 to 6 
 9 
 
 3 
 
 
 5 
 
 7 
 
 6 
 8 
 
 f 
 
 7 
 
 f 
 
 S 
 
 , 7 J 
 
 6 
 
 
 10 
 
 6 
 
 r 'I , 
 
 e 
 
 t 7 , 
 
 7 
 
 f 8 ' 
 
 8 
 . 1 , 
 
 10 
 
 9 
 
 6 
 
 i 
 
 1 
 
 
 
 , " . 
 
 to 
 
 /o 
 
The schemes after the delays and deletions are: 
 
 Page 25 
 
 
 , /3 
 
 IZ 
 
 ii 
 
 ID 
 
 
 IX 
 
 /3 
 
 '3 
 
 12. 
 
 IZ 
 
 II 
 
 
 
 II 
 
 /3 
 
 /0 
 /I 
 
 
 s 
 
 lo 
 
 
 10 
 
 IZ 
 
 7 
 
 
 , 1 
 
 
 7 
 
 1 i 
 
 8 
 
 7 
 
 3 
 
 k 
 
 
 f . 
 
 & 
 
 , S H . 
 
 > 
 
 9 
 
 (2 
 II 
 
 12 ? 
 /0 , 
 
 n 6 
 
 . 1 
 
 lo 2 
 
 1 
 
 IZ 
 
 
 10 . 
 
 3 
 , 1 
 
 
 10 
 
 , '3 , 
 
 
 e 
 
 r " 
 
 V l0 , 
 
 
 H 
 
 (3 
 
 JO 
 >Z 
 
 1 
 
 5- 
 , 10 . 
 
 
 12. 
 
 , !.? , 
 
 13 
 
 
 6> 
 
 10 
 
 
 
 > *J " X 
 
 , 8 
 
 . f 
 
 
 /» 
 
 // 
 
 6 
 
 9 «• 
 
 II 
 
 lo 
 
 
 . ? * 
 
 , 1 
 
 
 , '° 
 
 - '1 
 
 5 
 
 6 
 
 7 
 
 
 
 
 
 1 
 
 ,^ 
 
 1 
 
 
 /0 i 
 
 II , 
 
 V 
 
 5 
 
 6 
 
 
 ? 
 
 
 g 
 
 
 6 
 
 ■ f i 
 
 
 . 1 J 
 
 10 
 
 3 
 
 * 
 
 £• 
 
 
 6 
 
 ? 
 
 «>-*- 
 
 z 
 
 1 
 
 3 
 
 .. * ,,* , 
 
 4 
 
 a ' 
 
 . *■ 
 
 3 
 
 . 4 
 
 5" 
 
 
 
 y 
 
 3 
 5 
 
 f 4 
 
 B 5- 
 7" , 
 
 1 
 8 
 
 
 IO 
 
 3 
 
 It 
 
 
 6 
 
 ? 
 
 , * , 
 
 7 
 
 
 Y 
 
 5" 
 
 6 
 9 
 
 7 
 
 , 1 
 
 to 
 
 
 
 5- 
 
 4 
 
 1 
 
 8 , 
 
 8 
 
 ID 
 
 II 
 
 
 6 
 
 1 
 
 to 
 
 it 
 
 ID 
 
 i 
 
 lo 
 
 > 
 
 The time required by the above scheme is max{ t(m-i+l,n-j+l) , t(i,n- j+l)+l, 
 t(m-i+l, j)+3, t(i,j)+4} - max{m+n-i- j+1, n+i-j+1, m+j-i+3, i+j+3} which is 
 m+n-i-j+1 when m>2i and n>2j+2. 
 
 Schemes for other interior vertices may be constructed using a similar 
 technique. 
 
Page 26 
 
 4 Fault-tolerant Broadcasting In Wraparound Grids . 
 
 Let s(m,n) [i(m,n)] be the minimum time required for 1 fault-tolerant 
 
 broadcasting from a vertex of S [I ]. 
 
 m,n m,n 
 
 Theorem 4.1 : 
 
 s(m,n) ,i(m,n) > 
 
 [m/2] + [n/2] +1 if m or n is even; m,n>2 
 
 [m/2] + [n/2] 
 
 if m and n are odd; m,n>3 
 
 Proof: To complete a fault-tolerant broadcast in S or in I requires 
 r m,n m,n n 
 
 [m/2 J + [n/2 J time units if m or n is even and both are greater than 2 and 
 [m/2 J + [n/2 J + 1 time units if both m and n are odd and greater than 3 [Far- 
 ley & Hedetniemi, 78]. Clearly any scheme which completes 2 calling paths to 
 each vertex of either grid must take at least one time unit more than a 
 fault-tolerant scheme. 
 
 Theorem j4.2_: There is a calling scheme which completes 1 fault-tolerant broad- 
 cast from any vertex of S or I with m,n>4 in [m/2] + [n/2l + 1 time 
 
 m,n m,n ' L J L J 
 
 units. This is known to be minimum time if m or n is even. 
 
 Proof : Consider S . Without loss of generality m>n and the message origina- 
 tor Is ((m/2), (n/2) ). Let p=(m/2) and q=(n/2). The following scheme com- 
 pletes 1 fault-tolerant broadcast in [m/2j + [n/2] + 1 time units: 
 
 vertex 
 
 time 
 
 talks to 
 
 vertex 
 
 time 
 
 talks to 
 
 (1,1) 
 
 p+q 
 
 p+q+1 
 
 (2,1) 
 (l,n) 
 
 (1,2) 
 
 p+q 
 p+q+1 
 
 (2,2) 
 (1.3) 
 
 Cl.J) 
 
 2<j<q 
 
 p+q-j+2 
 p+q-j+3 
 
 P+q-j+4 
 
 (2,j) 
 (l.j+D 
 
 (l.j-D 
 
 q<j<n-l 
 
 p+j-q+1 
 p+j-q+2 
 p+j-q+3 
 
 (2,j) 
 
 (l.J-1) 
 
 U,j+1) 
 
 d,n-l) 
 
 p+n-q 
 p+n-q+1 
 
 (2,n-l) 
 d,n-2) 
 
 (l,n) 
 
 p+n-q 
 p+q+1 
 
 (2,n) 
 (1,1) 
 
 (1,1) 
 
 p+q-i+1 
 
 (1+1,1) 
 
 (1,2) 
 
 p+q-i+1 
 
 (1+1,2) 
 
Page 27 
 
 KKp 
 
 p+q-i+2 
 p+q-i+3 
 
 (1-1,1) 
 (l.n) 
 
 (l.j) 
 KKp 
 2<j<q 
 
 p+q-i- 
 p+q-i- 
 p+q-i- 
 p+q-i- 
 
 ■J+3 
 ■j+4 
 ■J+5. 
 ■J+6 
 
 (1+1. j) 
 (1-1, J) 
 (I,j+D 
 (I,j-D 
 
 (i,n-l) 
 KKp 
 
 n+p-q- 
 n+p-q- 
 n+p-q- 
 
 -i+1 
 ■i+2 
 -1+3 
 
 (1+1, n-1) 
 
 (1-1, n-1) 
 (i,n-2) 
 
 (P.D 
 
 q 
 
 q+1 
 q+2 
 q+3 
 q+4 
 
 
 (P,2) 
 
 (P+1,D 
 
 (p-l.D 
 
 (P,n) 
 
 (P,2) 
 
 (p.q) 
 
 1 
 2 
 3 
 
 4 
 
 
 (p,q+D 
 (p,q-D 
 
 (p+l,q) 
 
 (p-i,q) 
 
 (p.j) 
 
 q+2<j<n 
 
 j-q 
 j-q+i 
 
 J-q+2 
 
 j-q+3 
 
 n+q-j+3 
 n+q-j+4 
 
 (P,j-D 
 (p,j+D 
 (p+i,j) 
 (p-i,j) 
 (p,j+D 
 (p,j-D 
 
 (P,q-D 
 
 2 
 3 
 4 
 5 
 2q+l 
 
 
 (p,q) 
 (p,q-2) 
 
 (p+l,q-l) 
 (p-l,q-D 
 
 (p,q-2) 
 
 (1,1) 
 p<Km 
 
 q+i-p 
 
 q+l-p+1 
 
 q+i-p+2 
 
 (i-i,D 
 (i+i,D 
 
 (l,n) 
 
 (l.j) 
 p<i<m 
 
 2<j<q 
 
 i+q-p- 
 i+q-p- 
 i+q-p- 
 i+q-p- 
 
 •J+2 
 
 J+3 
 
 ■j+4 
 
 ■j+5 
 
 (1-1, j) 
 (1+1, j) 
 (I,j+D 
 (I,j-D 
 
 (l.q+D 
 p+KKra 
 
 i-p+2 
 i-p+3 
 i-p+4 
 i-p+5 
 
 
 (1-1, q+1) 
 (1+1, q+D 
 
 (i,q) 
 
 (i,q+2) 
 
 (i,n-l) 
 
 n+i-q- 
 
 ■P 
 
 (1-1, n-1) 
 
 KKp 
 
 p+q-i+2 
 
 (1-1,2) 
 
 
 p+q-i+3 
 
 (i,3) 
 
 (l.j) 
 
 q+J-i- 
 
 -p+2 
 
 (l+l,j) 
 
 KKp 
 
 q+j-i" 
 
 ■p+3 
 
 (1-1, j) 
 
 q<j<n-l 
 
 q+j-i- 
 
 ■p+4 
 
 (l,j-l) 
 
 
 q+j-i- 
 
 -p+5 
 
 (I.j+D 
 
 (l,n) 
 
 n+p-q- 
 
 ■i+1 
 
 (i+l,n) 
 
 KKp 
 
 n+p-q- 
 
 •i+2 
 
 (i-l,n) 
 
 
 p+q-i+3 
 
 (1,1) 
 
 (P,j) 
 
 q-j+i 
 
 
 (P,j+D 
 
 Kj<q-1 
 
 q-J+2 
 
 
 (P.j-1) 
 
 
 q-j+3 
 
 
 (P+I.j) 
 
 
 q-j+4 
 
 
 (p-l.j) 
 
 
 q+j+2 
 
 
 (P.j-D 
 
 
 q+j+3 
 
 
 (P.j+D 
 
 (p,q+D 
 
 1 
 
 
 (p.q) 
 
 
 2 
 
 
 (p,q+2) 
 
 
 3 
 
 
 (p+l,q+l) 
 
 
 4 
 
 
 (p-l,q+D 
 
 
 7 
 
 
 (p+l,q+l) 
 
 (P,n) 
 
 n-q 
 
 
 (p,n-l) 
 
 
 n-q+1 
 
 
 (p+l,n) 
 
 
 n-q+2 
 
 
 (p-l,n) 
 
 
 q+3 
 
 
 Cp.D 
 
 
 q+4 
 
 
 (p,n-l) 
 
 (p,q+2) 
 
 2 
 
 
 (p,q-D 
 
 
 3 
 
 
 (p,q+D 
 
 
 4 
 
 
 (p+i.q) 
 
 
 5 
 
 
 (p-i.q) 
 
 
 n+1 
 
 
 (p,q+D 
 
 (1,2) 
 
 q+i-p 
 
 
 (i-1,2) 
 
 p<Km 
 
 q+i-p+1 
 
 (1+1,2) 
 
 
 q+i-p+2 
 
 (1,3) 
 
 (p+q,q+l) 
 
 3 
 
 
 (p,q+D 
 
 
 4 
 
 
 (p+2,q+l) 
 
 
 5 
 
 
 (p+i,q) 
 
 
 6 
 
 
 (p+l,q+2) 
 
 
 7 
 
 
 (p,q+D 
 
 (i,j) 
 
 i+J-P" 
 
 •q+1 
 
 (1-1, j) 
 
 p<Km 
 
 1+J-P- 
 
 ■q+2 
 
 (i+l,j) 
 
 q+Kj<n-l 
 
 i+J-P" 
 
 ■q+3 
 
 (l,j-l) 
 
 
 l+j-p- 
 
 ■q+4 
 
 (l,j+l) 
 
 (i,n) 
 
 n+i-q-p 
 
 (1-1, n) 
 
Page 28 
 
 p<i<m 
 
 n+i-q-p+1 
 n+i-q-p+2 
 
 (i+l,n-l) 
 (i,n-2) 
 
 p<i<m 
 
 n+i-q-p+1 
 q+i-p+2 
 
 (i+l,n) 
 (i,D 
 
 (m,l) 
 
 q+m-p 
 q+m-p+1 
 
 (m-1,1) 
 (m,n) 
 
 (m,2) 
 
 q+m-p 
 q+m-p+1 
 
 (m-1,2) 
 (m,3) 
 
 (m,j) 
 
 2<j<q 
 
 q-j+m-p+2 
 q-j+m-p+3 
 q-j+m-p+4 
 
 (m-l,j) 
 (m,j+l) 
 (m,j-l) 
 
 (m,j) 
 q<j<n-l 
 
 j-q+m-p+1 
 j-q+m-p+2 
 j-q+m-p+3 
 
 (m-l.j) 
 (m,j-l) 
 (m,j+l) 
 
 (m,n-l) 
 
 n-q+m-p 
 n- q+m-p+1 
 
 (m-l,n-l) 
 (m,n-2) 
 
 (m,n) 
 
 n-q+m-p 
 q+m-p+1 
 
 (m-l,n) 
 (m,l) 
 
 Note that if n=5 the vertices (p,q+2) and (p,n) are identical. In this case 
 
 the calls given for (p,q+2) are deleted as is the call by (p,n) to (p,n-l) at 
 
 time q+4. The scheme is illustrated in the following example for S Q in 
 
 0,0 
 
 which unused edges are omitted: 
 
 CT f * 1 * * 
 
 8 8 
 
 c 
 
 JL_, D«? 
 
 \ * 
 
 c 
 
 
 
 
 2A 
 
 -i&- 
 
 5 s 
 
 5" 
 
 ? <f 
 
 ~D* 
 
 c: 
 
 t b 
 
 f 
 
 c 
 c 
 
 ? j 
 
 8 & 
 
 1- 
 
 a 
 
 s 
 
 e 
 — D 1 
 
Page 29 
 As In the schemes for simple grids the calling paths to most vertices 
 (i,j) are the sides of the rectangle with (p,q), (i,q), (i,j), and (p,j) as 
 the corners. For vertices in columns 1 and n the calling paths include the 
 edges in row p, one wraparound edge, and the edges in columns 1 and n between 
 rows p and i. The calling paths for vertices in row p (with the exception of 
 (p,q+l)) form the set of row p edges. The vertex (p,q+l) is informed by the 
 square of edges including the corners (p,q) and (p+l,q+l). This is to avoid 
 completing the second calling path by labelling the edge between (p,q+l) and 
 (p,q+2) with n+2 which is greater than [m/2 J+[n/2j+l when m=n and both are 
 even (as in the example above). 
 
 The wraparound edges used by the above scheme are those between columns 1 
 and n. The scheme assumes that the grid is oriented so that m>n. This scheme 
 
 may be applied directly to I if n>m. That is, after reorienting I the 
 
 3 KK J m,n ' 6 m,n 
 
 scheme may be used since it ignores the edges between rows 1 and n. If n<m in 
 
 I the scheme must be altered as follows: 
 m,n 
 
 1. Edges between (i+1,1) and (i,n) with Ki<m-1 are labelled 
 with the larger of the labels given to the edges (i,l), 
 (i,n) and (i+l,n), (i+l,n) in the above scheme. 
 
 2. (1,1) calls (m,n) at time p+q+1. 
 
 3. The calls between (p,j) and (p,j+l) with Kj<q-2 at time 
 q+j+3 are delayed until q+j+4. 
 
 4. There is no call between (p,q-l) and (p,q-2) at time 
 2q+l. 
 
 5. There is a call between (p,q-l) and (p+l,q-l) at time 8 
 (7 if q=3). 
 
 This scheme completes the proof for I 
 
 v r m,n 
 
Page 30 
 Corollary U_.2: The 1 fault-tolerant scheme above requires 2mn-2m-l total calls 
 as compared to mn-1 for the fault-tolerant scheme. 
 
 Theorem 4^: There is a calling scheme which completes 1 fault-tolerant broad- 
 casting on S . or I , with m>4 or on S. or I. with n>4 in minimum time. 
 m,4 m,4 4,n 4,n 
 
 Proof ; Consider S ,. Without loss of generality the message originator is 
 
 1 III y H 
 
 (fm/2),2). The following scheme informs all vertices of S . with m>4 by time 
 
 m, 4 
 
 [m/2J + 3 which is minimum by Theorem 4.1: 
 
 a 
 
 fl+Z 
 
 a»3 
 
 4tl 
 
 At3 
 
 H*l 
 
 c 
 
 a 
 
 . ai-3 '. 
 
 3 3 
 
 Q- 
 
 \p+-l 
 
 a 
 
 V>H 
 
 to+l 
 
 k>+Z 
 
 7)a+3 
 
 
 
 D^ti 
 
 
 This scheme may also be used on S, or on I, by reorienting them as above. 
 
 In S, ,, six time units are required to complete 1 fault-tolerant broad- 
 cast. This is easily seen since all informed vertices must be involved in 
 calls at times 1 through 4 in order to complete a single path to every vertex 
 by time 4. In particular the message originator must call each of its neigh- 
 bors. Thus at time 5 no second paths can be completed by a call from the mes- 
 sage originator. Only an even number of second paths can be completed at time 
 5 since each call completes two of them. Since 15 points require second paths 
 
Page 31 
 an additional call must be made at time 6 to complete the scheme. The follow- 
 ing scheme informs all vertices of S, , by t»6: 
 
 CH 
 
 c; 
 
 c; 
 
 .^ZD 
 
 Z)5 
 
 5 
 
 In order to use these schemes for I , the following changes must be made: 
 
 1. Edges between (1+1, 1 ) and (i,n) with Ki<m are labelled 
 with the larger of the two labels given to the edges (i,l), 
 (i,n) and (i+1), (i+l,n) in the above scheme. 
 
 2. (1,1) calls (m,n) at time [m/2]+3. 
 
 Corollary ^.3_: The 1 fault-tolerant scheme above requires mn+2m total calls as 
 compared to mn-1 for the fault-tolerant scheme. The scheme for S, , 
 requires mn+2m-l total calls. 
 
 Theorem f*»b} There is a calling scheme which completes 1 fault-tolerant broad- 
 casting from any vertex of S , or I with m>3 or S or I- with m>3 in 
 
 m,J m,J J,m J,m 
 
 time [m/2j + 3 if m is odd or [m/2] + 4 if m is even. 
 
 Proof : Consider S .. Without loss of generality the message originator is 
 
Page 32 
 
 (fm/2),2). The following schemes complete the proof for S .: 
 
 m, J 
 
 CZK^ p* 
 
 .+■>. 
 
 l+l 
 
 i+T- 
 
 a-H 
 
 c i *** i J_J 
 
 IO.+ 2. 
 
 (■.«*» 
 
 l«.+ i 
 
 c: 
 
 a-»-3 
 
 «.rX 
 
 ( J* 
 
 U 
 
 af2 
 
 , ar3 , 
 
 r A** r 
 
 , at3 J 
 
 a 
 1 *±2-h 
 
 
 4+3 , 
 
 a+4 
 
 
 a+L 
 
 <*-2 
 
 *H 
 
 
 
 . a+1 ,, d-rt , 
 
 
 a+3 a+4 '. 
 
 4 
 
 H 
 
 3 
 
 
 c 
 
 . z r 1 
 
 | * 
 
 3 
 
 < 
 3 
 
 2 
 
 
 
 . ao 
 
 . *H 
 
 
 a+3 a+H 
 
 
 at-Z 
 
 <U2 
 
 ■ 1 
 
 i — < 
 
 
 >+3 
 
 a = 
 
 /n oM 
 
 <*+3 at-f 
 m even 
 
 As above these schemes may be applied directly to S and I- . For 1 the 
 
 J,m J,m m, j 
 
 following changes must be made: 
 
 1. If m is odd label the edges between (i+1,1) and (i,n) for 
 Ki<[m/2j+2. 
 
 2. If m is even label the edges ([m/2j,l), ([m/2J-l,n) and 
 ([m/2),n), ([m/2]+l,l) with [m/2]+4. 
 
 Corollary f*_»b} The 1 fault-tolerant scheme above requires mn+2m total calls as 
 compared to mn-1 for the fault-tolerant scheme. 
 
 It should be noted that the wraparound grids are A edge-connected and 
 will admit 2 fault-tolerant and 3 fault-tolerant broadcast schemes [Liestman, 
 80]. Such schemes would include three and four calling paths to each vertex. 
 
Page 33 
 5 Summary * 
 
 The process of fault-tolerant broadcasting has been defined and investi- 
 gated in grid and grid-like graphs. The results show that the addition of 1 
 fault-tolerance to the broadcasting process requires a small increase in the 
 time required for broadcasting and a substantial increase in system utiliza- 
 tion. In particular, only a single time unit beyond the fault-tolerant 
 broadcast time is required to perform a 1 fault-tolerant broadcast on most 
 grids. The total number of calls required is roughly doubled when fault- 
 tolerance is added. 
 
Page 34 
 References. 
 
 [Cockayne & Hedetniemi, 78] Cockayne, E. J. and S. T. Hedetniemi, "A Conjec- 
 ture Concerning Broadcasting in M-dimensional Grid Graphs", Univ. of Ore- 
 gon Tech. Rept. No. CS-TR-78-14, 1978. 
 
 [Farley, 79] Farley, A. M. , "Minimal Broadcast Networks", Networks, Vol. 9, 
 No. 4, pp. 313-332, 1979. 
 
 [Farley & Hedetniemi, 78] Farley, A. M. and S. T. Hedetniemi, "Broadcasting in 
 Grid Graphs", Univ. of Oregon Tech. Rept. No. CS-TR-78-6, 1978. 
 
 [Farley, et al. , 79] Farley, A. M. , S. T. Hedetniemi, S. Mitchell and A. 
 Proskurowski , "Minimum Broadcast Graphs", Discrete Mathematics, Vol. 25, 
 pp. 189-193, 1979. 
 
 [Harary, 69] Harary, F. , Graph Theory , Addison-Wesley, Reading, Mass., 1969. 
 
 [Liestman, 80] Liestman, A., "Fault-tolerant Broadcast Graphs", Univ. of Illi- 
 nois Tech. Rept. No. UIUCDCS-R-80-1029, 1980. 
 
 [Mitchell & Hedetniemi, 78] Mitchell, S. L. and S. T. Hedetniemi, "A Census of 
 Minimum Broadcast Graphs", Univ. of Oregon Tech. Rept. No. CS-TR-78-7, 
 1978. 
 
 [Peck, 80] Peck, G. W. , "Optimal Spreading in an n Dimensional Rectilinear 
 Grid", Studies in Applied Mathematics, Vol. 62, No. 1, pp. 69-74, 1980. 
 
 [Van Scoy, 79] Van Scoy, F. L. , "Broadcasting a Small Number of Messages in a 
 Square Grid Graph", Proceedings of the Seventeenth Annual Allerton 
 Conference on Communication, Control and Computing, 1979. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-80-1030 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 4. Title and Subtitle 
 
 FAULT-TOLERANT GRID BROADCASTING 
 
 6. 
 
 7. Author(s) 
 
 Arthur L. Liestman 
 
 8. Performing Organization Rept. 
 
 No, R-80-1030 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois U-C 
 Urbana, IL 61801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSG 1471 
 
 12. Sponsoring Organization Name and Address 
 
 National Aeronautics and Space Administration 
 Hampton, Virginia 
 
 13. Type of Report & Period 
 Covered 
 
 Technical 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 Fault-tolerant broadcasting is simply broadcasting with redundancy in 
 order that certain faults, i.e. connections between nodes ceasing to 
 function, do not affect the completion of the broadcast. This paper 
 investigates the problem of fault-tolerant broadcasting in grid or 
 grid-like graphs. 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 fault- tolerant 
 
 broadcasting 
 
 grids 
 
 17b. Identifiers /Open-Ended Terms 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 unlimited 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 37 
 
 22. Price 
 
 FORM NTIS-35 (10-70) 
 
 USCOMM-DC 40329-P71