Minimum Average Delay of Routing Trees

The general communication tree embedding problem is the problem of mapping a set of communicating terminals, represented by a graph G, into the set of vertices of some physical network represented by a tree T . In the case where the vertices of G are mapped into the leaves of the host tree T the underlying tree is called a routing tree and if the internal vertices of T are forced to have degree 3, the host tree is known as layout tree. Different optimization problems have been studied in the class of communication tree problems such as well-known minimum edge dilation and minimum edge congestion problems. In this report we study the less investigate measure i.e. tree length, which is a representative for average edge dilation (communication delay) measure and also for average edge congestion measure. We show that finding a routing tree T for an arbitrary graph G with minimum tree length is an NP-Hard problem. 1 Definitions and Introductory Points Consider a group of terminals communicating via a finite network G = (V,E), where the set of vertices (finite set V ) and edges (finite setE), respectively represent the collection of terminals and their direct communication paths. We show ∣V ∣ by n and ∣E∣ as m. The general communication tree embedding problem is the problem of mapping the set of terminals into the set of vertices of some physical network represented by a tree T . Accordingly, the two vertices v, u ∈ V (G) that are directly connected via e ∈ E(G), are connected indirectly via some path PT (v, u) in T . In the case where the vertices of G are mapped to the leaves of the host tree, the underlying tree is called a routing tree. In this report we mostly focus on the case where the internal vertices of the host tree have degree 3 (known as tree layout problem). We denote the sets of leaf nodes and internal nodes of tree T respectively by VL(T ) and VI(T ). 1 For a graph G and a communication tree T for G, there are different measures defined in literature. In following we define the two measures that we are intrusted in this report. For a comprehensive list of measures, an interested reader can refer to [3]. Definition 1.1 (Edge Dilation). Consider a graphG and a communication tree T and a bijection φ from vertices of G to leaf nodes of T . The dilation λ(uv,T,φ,G) of an edge {u, v} ∈ E(G) is the distance between φ(u) and φ(v) in T . ◻ We represent the distance of two vertices {u, v} in a graph G with dG(u, v) Definition 1.2 (Edge Congestion). Give a graph G and a communication tree T and and a bijective mapping φ ∶ V (G) → VL(T ) . The congestion δ(xv,T,φ,G) and of an edge {x, y} ∈ E(T ) is the the number of edges in {u, v} ∈ E(G) that in T , the path PT (φ(v), φ(u)) traverse trough {x, y}. ◻ We try to use the term node in case of trees as opposed to the term vertex, which we use for general graphs. 1 ar X iv :1 60 1. 02 69 7v 1 [ cs .C C ] 1 2 Ja n 20 16 Based on the definition of the communication tree for a graph G, removal of every edge {x, y} ∈ E(T ) partitions the set of vertices of G into two component. Hence every edge of tree corresponds to a cut in G. Therefore the congestion of {x, y} ∈ E(T ) is the size of the cut it corresponds to. Several optimization problems can be defined based on these two measures. Minimum tree layout dilation is the problem of finding a tree layout for a given graph G such that the maximum edge dilation is minimized, where the maximum is taken over all edge of G. In [7] it is shown that the problem of finding a tree layout with minimum dilation is NP-hard, when the layout tree is rooted. Similarly, given a graph G, in minimum tree layout congestion problem the goal is to find a tree layout T , such that the maximum edge congestion is minimized. In [8] Seymour and Thomas show that the minimum tree layout congestion problem is polynomially solvable for the case of planer graphs, and is NP-hard when considering general graphs. In this report we study the minimum tree layout length problem (shortly called Min Tree Length), formally defined as it follows. Definition 1.3 (Minimum tree layout length). Consider the finite undirected graph G = (V,E). The minimum tree layout length problem is the problem of finding layout tree T and a bijective mapping φ ∶ V (G)→ VL(T ) such that L(T,φ,G) = ∑{u,v}∈E(G) λ(uv,T,φ,G) is minimized. ◻ It is not hard to see that ∑{u,v}∈E(G) λ(uv,T,φ,G) = ∑{x,y}∈E(T ) δ(xv,T,φ,G). Hence, in the rest of this report we may use them interchangeably. Accordingly, in the communication graph embedding problems, the dilation of an edge {u, v} ∈ E(G) abstractly represent the communication delay between vertices u and v. Similarly the congestion of an edge e ∈ E(T ) is a representative for the traffic on the physical link e. Hence tree length measure corresponds to the average delay between the vertices of G and also to the average edge congestion of the host tree. 2 Minimum Length of Tree Layout In the special case of tree layout problem, the underlying host graph is a tree T where the degree of every node is either 1 or 3 and the vertices of G are being mapped to leaves of T . In this section we study minimum tree layout length. We show that Min Tree Length problem is NP-hard for multi-graphs2, and later on we show the problem stays NP-hard when restricted to the class of simple graphs. 2.1 Min Tree Length of Complete Graphs Consider the complete graphG = (V,E) where ∀u, v ∈ V (G),{u, v} ∈ E(G). It is not hard to see that a layout tree T is a solution for the Min Tree Layout problem for G, iff ∣VL(T )∣ = n (and hence ∣V (T )∣ = 2n − 1), and the summation of distance of leaf nodes of T is minimized. We denote the summation of distances of leaves of a tree T by: σLL(T ) = 1 2 ∑ x,y∈VL(T ) dT (x, y) Leaf to leaf distance summation measure σLL is very similar to the definition of Wiener index (proposed by chemist Wiener [10]), which is the summation of distances of all vertices of a given graph G as represented in following equation. σ(G) = 1 2 ∑ u,v∈V (G) dG(u, v) By multi-graph we refer to finite graphs with possibility of parallel edges and no loop.