Optimal Bounds for the Change-Making Problem

The change-making problem is the problem of representing a given value with the fewest coins possible. We investigate the problem of determining whether the greedy algorithm produces an optimal representation of all amounts for a given set of coin denominations 1 = c_1 < c_2 < ... < c_m. Chang and Gill show that if the greedy algorithm is not always optimal, then there exists a counterexample x in the rangec_3 <= x < (c_m(c_m c_m-1 + c_m - 3c_m-1)) \ (c_m - c_m-1).To test for the existence of such a counterexample, Chang and Gill propose computing and comparing the greedy and optimal representations of all x in this range. In this paper we show that if a counterexample exists, then the smallest one lies in the range c_3 + 1 < x < c_m + c_m-1, and these bounds are tight. Moreover, we give a simple test for the existence of a counterexample that does not require the calculation of optimal representations.In addition, we give a complete characterization of three-coin systems and an efficient algorithm for all systems with a fixed number of coins. Finally, we show that a related problem is coNP-complete

A New Characterization of Tree Medians with Applications to Distributed Algorithms

A new characterization of tree medians is presented: we show that a vertex m is a median of a tree T with n vertices iff there exists a partition of the vertex set into [n/2] disjoint pairs (excluding m when n is odd), such that all the paths connecting the two vertices in any of the pairs pass through m. We show that in this case this sum is the largest possible among all such partitions, and we use this fact to discuss lower bounds on the message complexity of the distributed sorting problem. This lower bound implies that, given a network of a tree topology, choosing a median and then route all the information through it is the best possible strategy, in terms of worst-case number of messages sent during any execution of any distributed sorting algorithm. We also discuss the implications for networks of a general topology and for the distributed ranking problem

Graphs of Edge-Intersecting Non-Splitting Paths in a Tree: Representations of Holes-Part II

Given a tree and a set P of non-trivial simple paths on it, VPT(P) is the VPT graph (i.e. the vertex intersection graph) of the paths P, and EPT(P) is the EPT graph (i.e. the edge intersection graph) of P. These graphs have been extensively studied in the literature. Given two (edge) intersecting paths in a graph, their split vertices is the set of vertices having degree at least 3 in their union. A pair of (edge) intersecting paths is termed non-splitting if they do not have split vertices (namely if their union is a path). We define the graph ENPT(P) of edge intersecting non-splitting paths of a tree, termed the ENPT graph, as the graph having a vertex for each path in P, and an edge between every pair of vertices representing two paths that are both edge-intersecting and non-splitting. A graph G is an ENPT graph if there is a tree T and a set of paths P of T such that G=ENPT(P), and we say that is a representation of G. Our goal is to characterize the representation of chordless ENPT cycles (holes). To achieve this goal, we first assume that the EPT graph induced by the vertices of an ENPT hole is given. In [2] we introduce three assumptions (P1), (P2), (P3) defined on EPT, ENPT pairs of graphs. In the same study, we define two problems HamiltonianPairRec, P3-HamiltonianPairRec and characterize the representations of ENPT holes that satisfy (P1), (P2), (P3). In this work, we continue our work by relaxing these three assumptions one by one. We characterize the representations of ENPT holes satisfying (P3) by providing a polynomial-time algorithm to solve P3-HamiltonianPairRec. We also show that there does not exist a polynomial-time algorithm to solve HamiltonianPairRec, unless P=NP