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 UIUCDCS-R-78-948 
 
 UILU-ENG 78 1740 
 
 ON THE COMPLEXITY OF EDGE 
 LABELINGS FOR TREES 
 
 BY 
 
 S. Zaks and Y. Perl 
 
 November 1978 
 
 * 'J 
 
 i*5 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
 
 Univci&Hj - 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/oncomplexityofed948zaks 
 
ON THE COMPLEXITY OF EDGE 
 LABELINGS FOR TREES 
 
 S. Zaks" and Y. Perl ' 
 
 October 1978 
 
 : Department of Computer Science, University of Illinois at Urbana- 
 
 Champaign, Urbana, Illinois 61801. 
 
 The work of this author was supported in part by the National 
 
 Science Foundation under grant NSF MCS 77-22830. 
 ** On leave from the Department of Mathematics and Computer Science, 
 
 Bar- I Ian University, Ramat-Gan, Israel. 
 
ABSTRACT 
 
 Given a tree T with n edges and a set W of n weights, we deal 
 with labelings of the edges of T with weights from W, optimizing certain 
 objective functions. For some of these functions the optimization problem 
 is shown to be NP-complete (e.g., finding a labeling with minimal diameter), m 
 and for others we find polynomial-time algorithms (e.g., finding a labeling 
 with minimal average distance). 
 
TABLE OF CONTENTS 
 
 I. INTRODUCTION 1 
 
 II. RADIUS AND DIAMETER: NP-COMPLETE RESULTS 4 
 
 III. RADIUS: POLYNOMIAL RESULTS 9 
 
 IV. AVERAGE MEASUREMENTS: POLYNOMIAL RESULTS 13 
 
 V. OPEN PROBLEMS 20 
 
Wl 
 
 w 
 
 I. INTRODUCTION 
 
 1 . 1 Definitions 
 
 Lot T=(V,E) be a tree with vertices V={1 ,2,. . . ,n+l } and edqes 
 
 E={e-| , e ? ,..., e }. A vertex of degree one is a leaf , and an edge incident 
 
 th a leaf is a terminal edge . P denotes a path with n vertices. W={w,, 
 
 w> is a set of weiqhts such that 0<w . =w, <w < . ..< w =w 
 
 2 n ■ — mm 1—2— — n max 
 
 A labeling of the edges of T with the weights from W is a bijection f : E -*■ W. 
 In this paper we consider labelings that optimize certain objective functions. 
 Let p(i,j) denote the (unique) path connecting vertices i and j in 
 T. Given a labeling f of T, the distance d f (i,j) between i and j is 
 
 d f (i,j) = I f(e) . 
 
 eep(i,j) 
 
 The diameter D~[1) of the tree T, labeled with f, is 
 
 D f {l) = max (d f (i,j)} . 
 i<j 
 
 Note : We omit the references to f and T whenever possible; i.e, we use the 
 notations d(i,j) and D instead of d^(i,j) and DAT), respectively, and 
 similarly for the functions defined in the sequel. 
 
 A center of a labeled tree is a vertex i for which max {d(i,j)} 
 is minimal, and this value is called the radius R of the tree; i.e., 
 
 R = min {max {d(i ,j)}} . 
 i j 
 
 The terms diameter, radius and center are similarly defined for labeled 
 
 undirected graphs. In a rooted tree the center is located at the root. 
 
 The maximal weight and minimal weight on p(i ,j) are 
 
 max(i.j) = max (f(e)} and min(i,j)= min (f(e)> , 
 eep(i.j) eep(i.j) 
 
 respectively. 
 
p(i,j) is the number of edges in p(i,j). 
 
 is the average weight on p(i,j). 
 1 .2 The Objective Functions 
 
 The quantities which we optimize, for a given tree T, over all 
 possible labelings f, are the following: 
 
 1. 
 
 D , 
 
 
 
 
 2. 
 
 R , 
 
 
 
 
 3. 
 
 SUMDIS(J) 
 
 = z 
 
 d(i,j) , 
 
 
 4. 
 
 SUMMAX[J) 
 
 KJ 
 
 = I 
 
 max(i,j) , 
 
 
 5. 
 
 SUMMIN{T) 
 
 = Z 
 
 min(i,j) , 
 
 and 
 
 6. 
 
 sumavb(i) 
 
 KJ 
 
 = z 
 
 avr(i,j) . 
 
 
 The functions (3)-(6) are average measurements ,si nee the summations are 
 over all pairs of vertices. 
 1 .3 Motivation 
 
 This work is applicable in the design of communication networks. 
 Given communication lines of different properties, such as communication 
 cost, capacity, vulnerability and reliability, we want to assign these comm- 
 unication lines to the direct connections ("edges") of a given communication 
 network (of a tree structure), such that certain objective cost functions 
 will be optimized. 
 
 For example, the function SUMDIS is measuring the average communi- 
 cation cost of the network (see [4] and [8]). The diameter is measuring 
 the maximal communication cost in the network, and the radius is measuring 
 the maximal communication cost from a directory optimally located in the 
 
network. Let c(e) and v(e) be the capacity and vulnerability of a communi- 
 cation line assigned to the edge e, respectively. Define the capacity and 
 vulnerability between the vertices i and j as 
 
 c(i,j) = min (c(e)} and v(i,j) = max (v(e)} , 
 eep(i.j) eep(i,j) 
 
 respectively. Then the functions SUMMIT and SUMMAX measure the average 
 
 capacity and vulnerability of the network, respectively. 
 
 1 .4 In section 2 we present several NP-complete problems, concerning 
 
 optimizing the radius and the diameter of a tree and of a binary weighted 
 
 (0,1 weights) general graph. Polynomial-time algorithms for special cases 
 
 of minimizing the radius of a tree are shown in section 3. In section 4 
 
 we present polynomial -time algorithms for optimizing average measurement 
 
 functions of the tree, such as SUMDIS and SUMMAX. Open problems are found 
 
 in section 5. 
 
II. RADIUS AND DIAMETER: NP-COMPLETE RESULTS 
 
 2.1 In this section the following problems are shown to be NP-complete (see 
 [6]): 
 
 PROBLEM 1 : Given a tree T with n edges, a set W of n non-negative 
 
 weights, and an integer k>0, to determine whether there 
 
 exists a labeling f with diameter <_ k. 
 PROBLEM 2 : Like PROBLEM 1, for radius < k. 
 Note :! .This result holds also for rooted trees. 
 
 2. PROBLEMS 1 and 2 remain NP-complete when the weights are bounded 
 PROBLEM 3 : Like PROBLEM 1, for radius > k. 
 Note_:l. Maximizing the diameter of a tree is trivial. 
 
 2. PROBLEMS 1 , 2 and 3 are NP-complete also for the corresponding 
 vertex labeling problems. 
 PROBLEM 4 : Given a connected graph G with m edges and a set of 0,1 
 
 weights W, |W|=m, to determine whether there exists a 
 
 labeling with radius <_ 1. 
 PROBLEM 5 : Given a connected graph G with n+1 vertices and a set of 
 
 0,1 weights W, |W|=n, to determine whether G contains a 
 
 spanning tree which can be labeled such that its diameter 
 
 < 2. 
 
 2.2 The reductions are from the PARTITION problem and the MAXIMUM TERMINAL 
 SPANNING TREE problem, both known to be NP-complete (see [6] and [2] ): 
 
 PARTITION : Given positive integers a i , i=l,2,...,n, to determine 
 
 whether there exists a set Ic{l ,2, . . . ,n} such that 
 
 E a. = i a- . 
 iel HI 
 
 MAXIMUM TERMINAL SPANNING TREE : Given a graph G and an integer k>0, 
 
 to determine whether G has a spanning tree with at least 
 
 k terminals (=leaves). 
 
2.3 Proof for PROBLEM 1 
 
 We show that PARTITION is reduced to PROBLEM 1. Given a., 1-1 ,2,..., n, 
 (an instance of PARTITION), we define the following instance of PROBLEM 1: 
 
 T : 
 
 ~^ n edges *\^ 
 
 W = {a-,, a«, ..., a. 0, 0, z. a.} . 
 
 c. n v. ~, j 
 
 n O's 
 
 k " I Z i a i ' 
 
 We prove that there exists a solution to the PARTITION problem iff there exists 
 
 a labeling of T with diameter <_ k. 
 
 Suppose there exists a solution to the PARTITION problem, namely 
 
 z a. - z a. for some l£{l,2,...,n} . The following labeling has a 
 iel n i*fl n 
 diameter k: label AB with z. a. , spread the a.'s for iel and i^I on 
 
 BC and BD, respectively, and label the rest of the edges with O's. 
 
 Suppose there exists a labeling with diameter <_ k. One can find 
 such a labeling in which za. labels the edge AB (if in the given labeling 
 AB is labeled with a., for some t, and za. labels another edge e, then by 
 interchanging za. and a. the diameter is not increased). This means that the 
 a.'s on BC and on BD must sum up to j Za. each, hence we have a solution 
 to the given PARTITION instance. □ 
 2.4 Proof for PROBLEM 2 
 
 We prove that PARTITION is reduced to PROBLEM 2. Given a. (as above), 
 we define the following instance of PROBLEM 2: 
 
T ■ P 2n + 1 
 
 M = {a,, a,, ..., a„, 0, 0} , and 
 
 k = lza n0 ' s 
 
 k 2 za i . 
 
 The rest follows immediately. rj 
 
 Note : The same reduction holds in the case when this path is rooted in its 
 
 center, which proves that PROBLEM 2 is NP-complete also for rooted trees. 
 
 Note : As was pointed out by Dave Johnson [5], PROBLEM 1 and PROBLEM 2 
 
 remain NP-complete when the weights are bounded, and this is shown by a 
 
 similar reduction from 3-PARTITION ( see [2]), ( "NP-complete in the strong 
 
 sense" ). 
 
 2.5 Proof for PROBLEM 3 
 
 We prove that PARTITION is reduced to PROBLEM 3. Given a. (as above), 
 we define the following instance of PROBLEM 3: 
 
 T ' P n + 2 • 
 
 W = {a-. , a«» . .. i a , x} where x>max {a.} , and 
 
 k = x + p- «za. 
 
 If there exists a solution to PARTITION then the labeling shown in 
 figure 1 has a radius R = k. 
 
 / 
 
 a.'s for iel ^x a.'s for i^I 
 
 Figure 1 
 
 If there exists a labeling with radius /?>_ x + j Ea., then let D be 
 a center and DB a radius (see figure 2). Then 
 
 d(D,B) >_ x + 1 za. , and hence d(A,D) <_ j la. . 
 
A D y C B 
 
 • • • • • • • • • • • • 
 
 Figure 2 
 Also d(D,B) <_d(A,C), otherwise D is not a center. Therefore 
 
 x + \ sa -j 1 d(D,B) <_ d(A,C) < y + d(A,D) < y + j Za i , 
 hence x<y. But y<x, hence y=x, and 
 
 d(A,D) = d(B,C) = j 2a. , 
 
 which solves the PARTITION problem. □ 
 
 2.6 Note 
 
 PROBLEM 1, PROBLEM 2 and PROBLEM 3 are also NP-complete for vertex 
 labeling. The reductions are from the corresponding edge labeling problems. 
 Adding a weight to the given weights, a solution for the vertex labeling 
 problem induces a solution for the corresponding edge labeling problem. This 
 is done by regarding the vertex labeled as a root of the tree and using a 
 label of any other vertex to label the edge directed to this vertex. 
 
 2.7 Proof for PROBLEM 4 
 
 We prove that MAXIMUM TERMINAL SPANNING TREE is reduced to PROBLEM 4, 
 Given a graph G with n+1 vertices and k>0 (an instance of MAXIMUM TERMINAL 
 SPANNING TREE), we define the following instance of PROBLEM 4: G is the same 
 given graph and W contains n-k O's and all the rest l's. We prove that G has 
 a spanning tree with at least k leaves iff G has a labeling with radius /?<_! . 
 
 Suppose G has a spanning tree with at least k leaves. This yields 
 a labeling of G with radius R<_ 1 , by labeling the internal edges (of the tree) 
 with O's , and thus ewery internal vertex (of the tree) is a center for the 
 graph G with radius i?<l . 
 
Suppose there exists a labeling of G with radius R<_ 1. Let A be a 
 center of the labeled graph, and apply a shortest-path algorithm from A to get 
 a "shortest-path spanning tree" with radius R<_ 1. The tree is labeled with at 
 least k l's. In this tree there exists at most one edge labeled 1 on each path 
 from A to any leaf, since the radius is <_ 1. Therefore the number of leaves in 
 the spanning tree is at least as the number of l's in the tree, i.e. >_ k. □ 
 
 2.8 Proof for PROBLEM 5 
 
 The reduction is similar to that of PROBLEM 4. D 
 
 2.9 Conjecture 
 
 The following problem is also NP-complete: given a connected graph 
 and a set of 0,1 weights, to determine whether there exists a labeling with 
 diameter D<\ . 
 
 We have the following observations about this problem. Suppose the 
 graph G is labeled with the given weights such that its diameter D<\ . It is 
 clear that D=0 iff G contains a spanning tree with all edges labeled 0. Thus 
 the interesting case is when D=1. In this case we may assume that the edges 
 labeled with generate a forest in G, since if they generate any cycle we can 
 replace one 0-edge with a 1-edge without increasing the diameter (putting 
 this 0-edge in another arbitrary place not closing a cycle of 0's). Let T.= 
 (V.,E.) S i=l,2,...,j be the trees in this forest. A graph G'=(V , ,E') is 
 constructed from G by contructing (see [3] ) all the vertices of each tree T. , 
 i = l,2,...j, into one vertex u • . It can be shown that G is labeled with a 
 diameter z?=l iff G' is a complete graph. This observation lead us to the 
 above conjecture. 
 
III. RADIUS: POLYNOMIAL RESULTS 
 
 3.1 In the previous section we proved that minimizing the diameter or the 
 radius in a tree are NP-complete problems. Here we present polynomial -time 
 algorithms for certain instances of these problems. 
 
 3.2 PROBLEM 6 : Given a tree T and a set of 0,1 weights W, to find a labeling 
 
 with minimal radius. 
 The following algorithm solves this problem in 0(n) running-time: 
 Algorithm A 
 
 1. wh i 1 e (number of leaves in T) <_ (number of l's in W) 
 
 do begin 
 
 label all the terminal edges with 1 ; 
 delete the terminal edges from T ; 
 delete the used l's from W ; 
 end ; 
 
 2. Put all the remaining l's on terminal edges ; 
 label all other edges with 0's. 
 
 Proof : Left to the reader. □ 
 
 Note : Algorithm A is applicable also for rooted trees. 
 
 3.3 PROBLEM 7 : Like PROBLEM 6, for a set of a,b weights, a<b. 
 
 The following algorithm solves this problem in O(nlogn) running- 
 time in the case when T is a rooted tree. An 0(n logn) algorithm for free 
 trees is obtained by applying this algorithm from eyery vertex in the tree. 
 Algorithm B 
 
 1. Label all edges of T with a' s ; 
 
 insert all terminal edges into a "candidate list" ; 
 
 2. while W contains at least one b 
 
10 
 
 do begin 
 
 find an edge (x,y) in the "candidate list", which 
 lies on a path of minimal length from the 
 root to a leaf ; 
 label (x,y) with b ; 
 delete this b from W ; 
 delete (x,y) from the "candidate list" ; 
 if x has no sons labeled with a 
 then begin 
 
 find the unique w such that (w,x)eT ; 
 insert (w,x) into the "candidate list" ; 
 end ; 
 end ; 
 
 Proof : Let T be the tree labeled by the algorithm while V is an optimally 
 labeled tree minimizing the radius and satisfying the condition that if a 
 (directed) edge (x,y) is labeled with b then all edges in the subtree T' 
 rooted at y are labeled with b (all the b-weights are pushed towards the 
 leaves). Note that T satisfies this condition by the algorithm. 
 
 First we show that there exists such an optimal labeled tree T 1 . 
 Let T" be an optimal labeled tree not satisfying this property. Denote by 
 d, d' and d" distances in T, T 1 and T", respectively (note that all these 
 three trees differ only by their labelings). There exists an edge (x,y) 
 labeled b in T" while some edge (w,z) in T" is labeled with a. Exchanging 
 the weights of the edges (x,y) and (w,z) yields a labeling where d"(r,u) is 
 decreased by b-a for every ueT"-T" and is not changed for any other vertex 
 in the tree. Repeating this process yields a labeled tree T 1 , satisfying 
 the desired property, without increasing the radius, and therefore T is also 
 optimal. 
 
11 
 
 Assume now that the tree T labeled by the algorithm is not optimal, 
 i.e., there exists a vertex xeT such that d(r,x)>/?, where R is the radius 
 of T' . Then the path from r to x contains an edge labeled with b in T and 
 labeled with a in T' . Let (y,z) be the first such edge from the root r. On 
 the other hand there exists an edge (u,v) labeled with b in T' and labeled 
 with a in T. By the property of T all the edges in the subtree T' are 
 labeled with b. Let w be a vertex in T' with maximal distance d'(r,w) . 
 d'(r,w)<i? since R is the radius of T' . (see figure 3). 
 
 Figure 3 
 
 But in T 
 
 d(r,w) <_d'(r,w) + (a-b) <_ R + (a-b) 
 
 since the edge (u,v) is labeled with a in T. On the other hand, before the 
 algorithm labeled (y,z) in T with b, we had 
 
 d(r,x) > R + (a-b) . 
 
 Hence the algorithm should have labeled the edge (u,v) with b before labeling 
 
12 
 
 the edge (y,z), a contradiction. 
 
 Therefore the labeling of T is optimal. □ 
 
 3.4 PROBLEM 8 : Given a tree T and a set of 0,1 weights W, to find a labeling 
 
 with minimal diameter. 
 This is done in 0(n) running-time algorithm, identical to that of 
 PROBLEM 6. 
 
 3.5 PROBLEM 9 : Like PROBLEM 8, for a bounded number of distinct weights. 
 
 Also in this case the solution is polynomial. This is done in a 
 dynamic programmwng technique, proposed by Nimrod Megiddo [9] , and is similar 
 to the one described in the sequel (see PROBLEM 13, section 4.9). The details 
 are left to the reader. 
 
13 
 
 IV. AVERAGE MEASUREMENTS: POLYNOMIAL RESULTS 
 
 4.1 In this section we present polynomial -time algorithms for optimization 
 problems concerning the functions SUMDIS , SUMAVR, SUMMIN and SUMMAX. 
 
 4.2 PROBLEM 10 : Given T and W, to find a labeling that maximizes or 
 
 minimizes SUMDIS{1) = z d(i,j) . 
 
 i<j 
 
 Following T. C. Hu [4] we observe that instead of summing the 
 d(i,j)'s we can count the number of occurrences of each edge in the summation. 
 Suppose that an edge e partitions the tree T into two subtrees with k and n+l-k 
 vertices, respectively. Then e is counted k(n+l-k) times in SUMDIS{1). We 
 call this value c(e) = k(n+l-k) the connection of e. Now 
 
 sumdis (1) = z c(e)-f(e) 
 e 
 
 and the next theorem follows: 
 
 Theorem : Given the connections of the edges of the tree T, c(e-,) <_ c(e 2 ) ± ••• 
 
 <_ c(e ) , the labeling of e. with w. ( w n+ -i ,•)> f° r i=l ,2,... ,n, 
 
 maximizes (minimizes) the function SUMDIS '(T) . 
 
 The optimal labeling is unique up to permuting edges with equal 
 connections. Assuming that all weights are distinct, the number N of optimal 
 labelings satisfies 
 
 2 UJ < N < n! 
 
 The bounds are achieved for the graphs P ,, and K-, 
 
 3 r n+1 i ,n 
 
 The connections can be calculated in 0(n) time by regarding T as 
 rooted and traversing it in post-order (see [6] ). Therefore, and because of 
 the sorting step involved, the algorithm for optimal labeling (for either 
 maximizing or minimizing SUMDIS{1) ) has O(nlogn) running-time. 
 
14 
 
 4.3 PROBLEM 11 : Given T and W, to find a labeling that maximizes or 
 
 minimizes SUMAVR{1) = z i J 9 N , . 
 
 i<: j |p(i.j)| 
 
 The solution is essentially similar to the solution of PROBLEM 19. 
 The only change is that the connections c(e) are replaced by the average 
 connection c(e) defined by 
 
 c(e) = z c.(e)/j , 
 J J 
 
 where c.(e) is the number of paths of length j through the edge e. 
 
 J 
 
 Calculating the average connections is done by a breadth-first search 
 (see [1] ) from every edge e, getting the number of paths of length i on each 
 side of e (for every i), and then "merging" the information from both sides of 
 e by convolution doable in O(nlogn) time (see [1] ). Therefore the algorithm 
 in this case is of 0(n logn) time. 
 
 4.4 PROBLEM 12 : Given T and W, to find a labeling that minimizes 
 
 SUMMAX{1) = I max(i ,j) . 
 
 The problem of maximizing SUMMAX[J) is still open. Similarly, our 
 algorithm for solving PROBLEM 12 solves also the problem of maximizing the 
 function SUMMIN{1) , while the problem of minimizing SUMINiJ) is still open. 
 Conjecture : The problem of maximizing (minimizing) SUMMAX(J) ( SUMMIN(l) ) 
 is NP-hard. 
 
 The algorithm to solve PROBLEM 12 is based on the following theorem: 
 Theorem : Given T and W as above, and assuming w. < w. +1 for every i, we have: 
 
 1. In any labeling that minimizes SUMMAX(J) the largest weight 
 
 w labels a terminal edge, 
 n 3 
 
 2. For any terminal edge there exists an optimal labeling in which 
 
 w labels this edge, 
 n 3 
 
 3. min {SUMMAX f (J)} = ? i. w . (which means that this optimal 
 
15 
 
 value is independent on the tree T). 
 Note : The result holds also for the case <_ w, <_ ••• < w , and the modi- 
 fication of the proof is left to the reader. 
 
 Proof : We prove the theorem by induction on n. The theorem holds for n=l ,2 
 and 3 . Assume it holds for a tree with less than n edges. Let T be a tree 
 with n edges. Assume f is a labeling that minimizes SUMMAX(1) , and that w 
 labels a non-terminal edge e, thus partitioning T into two subtrees T, and Tp 
 containing k and n+l-k vertices, respectively, for some 2 <_ k £n-l. Wlog 
 we assume that w _, labels an edge in T, . Applying the inductive hypothesis 
 to T, we know that there exists an optimal labeling for T, in which a specified 
 terminal edge is labeled with w ,. We choose this edge to be such that by 
 removing it with its corresponding leaf we won't disconnect T (see figure 4). 
 
 k vertices 
 
 n+l-k vertices 
 
 Figure 4 
 
 Note that all the paths from vertices in T, , containing w , are not effected 
 by this new labeling of T, (since on them w is maximal), and, as for T, , the 
 value of SUMMAX is not changed by the inductive hypothesis. In the current 
 labeling the contribution of w and w _, to SUMMAX(T) is 
 
 k(n+l-k)w n + (k-l)w n _ 1 
 
 (1) 
 
 By Interchanging w .j and w we get a labeling f in which their contribution 
 is 
 
16 
 
 nw n + (k-l)(n+l-k)w n _ 1 . (2) 
 
 But substracting (1) from (2) gives 
 
 (2) - (1) = (k-l)(n-k)(w n _ 1 - w n ) < 
 
 because l<k<n and w _,<w , a contradiction. 
 
 We know that in an optimal labeling of T w labels a terminal edge 
 It is clear that the contribution of w to SUMMAX(T) is 
 
 nw (3) 
 
 n v ' 
 
 It is also clear that we can label any terminal edge with w , and, after 
 erasing it together with the corresponding leaf, we apply the inductive 
 hypothesis to the remaining tree, getting optimal labeling with a value of 
 
 n-1 
 
 £ iw. , 
 i = l ^ 
 
 n 
 hence for T we get the optimal value of I iw. . D 
 
 i = l n 
 
 Following the above proof, it is clear how to get an 0(n) labeling 
 
 algorithm that minimizes SUMMAX{1) . 
 
 4.5 As for the number of optimal labelings, it can be shown that, when all 
 the weights are distict, the number N of such labelings satisfies 
 
 2 <_ N <_ n! , 
 
 when the bounds are achieved for the graphs P , and K, . 
 
 4.6 PROBLEM 13 : Given T and W, to find a labeling that maximizes 
 
 SUMMAX(J). 
 We stated our conjecture (above) that this problem is NP-hard. As 
 a matter of fact, we don't even know how to solve this problem in the case 
 
17 
 
 when 
 
 the tree T is a path (P + , ) . Note that the "natural" approach of labeling 
 
 the edge with maximal connection with the largest weight w is not optimal; 
 for example, consider the case when T=P g and W={1 ,2,3,8,9} . 
 
 4.7 We consider now the case when T is a path and W contains a bounded number 
 of distinct weights. We present an exact description of the solution when W 
 contains only two kinds of weights, and give an optimal labeling algorithm 
 for the case when W contains a bounded number of distinct weights. 
 
 4.8 A Path, 2 Weights 
 
 Here T=P + , , and W contains k weights b and n-k weights a, a<b, 
 The structure of the solution is characterized in the following theorem: 
 Theorem : Let T and W be as above, and denote a labeling f as a sequence 
 
 x n x i x t-i x + 
 a u ba 'b-.-ba z 'ba z (4) 
 
 where x. ^0 and ex. = n-k. If the labeling f is optimal, 
 then 
 
 l x i " x -j I 1 1 f° r every 1 and j. 
 
 Proof : For a labeling function f as described in (4), Let N f (a) (N f (b)) 
 denote the number of paths p(i,j) in which the largest label is a (b). It 
 is clear that by minimizing N f (a) over all possible labelings f we will also 
 minimize SUMMAX[T) over all possible labelings f, since a<b and 
 
 N f (a) + N f (b) = ("+ 1 ) . 
 
 It is clear that 
 
 M-e°; •(":) m » 
 
 Assume that f is an optimal labeling and that the theorem doesn't hold. Wlog 
 we may assume that 
 
 x Q > x^l . (6) 
 
18 
 
 We define a labeling f as follows: 
 
 Art ~ I X -i "■" I Art Al-i X » 
 
 f : a u ba ' ba ^b ••• ba r "'ba r (7) 
 
 By (5) we have 
 
 "<-«•(-':) (■•;)- <•> 
 
 Using (5), (8) and (6) it can be shown that 
 
 M f ,(a) < M f (a) , 
 
 hence SUMMAX fl {J) > SUMMAxAl) , a contradiction. D 
 
 Using this theorem it is clear how one should "spread" the a's among 
 
 the b's in order to get a labeling that maximizes the function SUMMAX. 
 
 4.9 A Path, k Weights 
 
 Here T=P + , , and W contains m. copies of w. for i = l,2,...,k, 
 
 where m, + m ? + ••• + m. = n and w, > w ? > ••• > w, >_ 0. The following 
 
 dynamic programming approach, proposed by Nimrod Megiddo [9] , solves the 
 
 2k+l 
 problem in 0(n ). Let T be a path of length is. labeled with s. copies 
 
 of w. for i=l,2,...,k, where the leftmost label is w and p. is the 
 i a l 
 
 position of the first occurrence of w. (from the left). Denote by 
 
 <t> ( s^, s 2 , ... , s k , w g , p-j, p 2 , ... , p k ) 
 
 the maximal value for SUMMAX(T) . Note that p =1. 
 
 a 
 
 If there exists b such that p,=2 then 
 
 *(s 1 ,...,s k ,w a ,p 1 ,...p k ) = max {<j>(sl , . . . ,s a _ 1 ,s a -l ,s a+] , . . . ,s k ,w b ,p^ ,. . . ,p k ) + 
 
 p a 
 
 *(a) } 
 
 where p! = p.-l for i^b and i^(a) is the contribution of the leftmost vertex 
 
19 
 
 vertex to SUMMAX(T) } easily calculated using the p-'s. If there exists 
 
 no such b then w labels the second edge too, and we compute <f> by 
 
 a 
 
 <fr(s 1 ,s 2 ,...,s k ,w a ,p 1 ,...,p k ) = ♦(s-j,...,s a _ 1 ,s a -l,s a+1 ,...,s k ,w a ,p 1 -l,...,p k -l) + 
 
 where ^(a) is as above. 
 
 The number of choices for the s.'s is 0(n ), and so is that for the 
 
 p. 's. Hence the function <|» is computed in 0(n ) points, where each calcu- 
 
 2k+l 
 lation is at most 0(n), therefore the algorithm runs in time 0(n ). 
 
20 
 
 V. OPEN PROBLEMS 
 
 We mention three open problems which arose while working on this 
 paper: 
 
 1. (see 2.9) Given a graph and a set of 0,1 weights, to determine 
 whether there exists a labeling with diameter <_ 1. 
 
 2. (see 4.4 and 4.6) Given a tree T=P + , (a path) and W, to find a 
 labeling which maximizes SUMMAX(l) . 
 
 3. Given a tree T and a set W of a,b weights, to find a labeling 
 which maximizes SUMMAX{J) (this is an interesting special case 
 of the open problem of maximizing SUMMAX(l) for general T and W) 
 
 ACKNOWLEDGEMENT 
 
 We would like to thank C. L. Liu for the inspiring conversations 
 we had while working on this paper. 
 
21 
 
 REFERENCES 
 
 [1] A. V. Aho, J. E. Hopcroft and J. D. Ullman, The Design and Analysis of 
 Computer Algorithms , Addison-Wesley, Reading, Mass. 1974. 
 
 [2] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide 
 to the Theory of NP-Completeness , W. H. Freeman and Company, San 
 Francisco, to appear. 
 
 [3] F. Harary, Graph Theory , Addison-Wesley, Reading, Mass. 1969. 
 
 [4] T. C. Hu, "Optimal Communication Spanning Trees," SIAM J. on Computing 3, 
 
 No. 3, September 1974, 188-195. 
 [5] D. S. Johnson, Private communication. 
 [6] R. M. Karp, "Reducibili ty among Combinatorial Problems," in Complexity of 
 
 Computer Computations , R. E. Miller and J. W. Thatcher (eds.), Plenum 
 
 Press, N. Y. 1972 , 85-104. 
 [7] D. E. Knuth, The Art of Computer Programming , Vol. 1: Fundamental 
 
 Algorithms, Addison-Wesley, Reading, Mass. 1968. 
 [8] J. K. Lenstra, A. H. G. Rinnooy Kan and D. S. Johnson, "The Complexity 
 
 of the Network Design Problem," Mathematisch Centrum, Amsterdam 1976. 
 [9] N. Megiddo, Private communication. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 UIUCDCS-R-78-948 
 
 3. Recipient's Accession No. 
 
 4. Tit If and Subi ii It- 
 
 5. Report Date 
 
 November 1978 
 
 ON THE COMPLEXITY OF EDGE LABELINGS FOR TREES 
 
 7. Authot(s) 
 
 S. Zaks and Y. Perl 
 
 8. Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 222 Digital Computer Lab 
 Dept. of Computer Science 
 University of Illinois 
 
 10. Project/Task/Worlc Unit No. 
 
 university or 1111 
 Urbana. IL 61801 
 
 11. Contract/Grant No. 
 
 NSF MCS 77-22830 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D.C. 
 
 13. Type of Report & Period 
 Covered 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 Given a tree T with n edges and a set W of n weights, we deal 
 with labelings of the edges of T with weights from W, optimizing certain 
 objective functions. For some of these functions the optimization problem 
 is shown to be NP-complete (e.g., finding a labeling with minimal diameter), 
 m and for others we find polynomial-time algorithms (e.g., finding a 
 labeling with minimal average distance) . 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 Tree, rooted tree, radius, diameter, up-complete, 
 
 17b. Identifiers/Open-Ended Terms 
 
 17c. COSATI Field/Group 
 
 18. Availability Statement 
 
 FORM NTIS-35 ( 10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 22. Price 
 
 USCOMM-DC 40329-P7 1 
 
rro P 4 
 
FEB 2fl MM