Minimum Spanning Trees. Lecture II: Minimium Spanning Tree Algorithms. An Idea. Some Terminology. Dr Kieran T. Herley

Transcription

1 Lecture II: Minimium Spanning Tree Algorithms Dr Kieran T. Herley Department of Computer Science University College Cork April 016 Spanning Tree tree formed from graph edges that touches every node (e.g. heavy edges above) Minimum Spanning Tree (MST) spanning tree with minimum total weight tree weight = sum of edge weights MST need not be unique KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April 016 / 5 Some Terminology An Idea A cut partition of V into two sets S and V S Edge (u, v) crosses cut if one endpoint is in S, the other in V S Cut respects set A E if no edge in A crosses the cut Set A E is extendible if A is a subset of some MST. Edge (u, v) A is safe for A if A {(u, v)} is also extendible. Create MST by growing it edge by edge: Algorithm GenericMST(G): Create empty set A while A does not span all nodes do find (u, v) safe for A add (u, v) to A return A KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April 016 / 5

2 Finding Safe Edges Let Nodes(A) = nodes spanned by A. Proof of Claim Claim For extendible A, the cheapest edge (u, v) crossing cut (Nodes(A), V Nodes(A)) is safe for A. Suppose S containing A is an MST that does not include (u, v). Claim Cheapest graph edge is safe for A =. Claim For extendible A, the cheapest edge (u, v) crossing cut (Nodes(A), V Nodes(A)) is safe for A. KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 S (u, v) contains a cycle. Some (x, y) S also crosses cut. weight(u, v) weight(u, v) by choice of (u, v). Thus, S minus (x, y) plus (u, v) contains A spans V has min. weight KH (1/10/16) Lecture II: Minimium Spanning Tree i.e. Algorithms (u, v) is safe forapril A / 5 Kruskal s algorithm: the basic idea K s algorithm b c 7 d Start with forest of n one-node trees Sort edges in non-increasing order. Take each edge e in turn: if e bridges two different trees, mark e (and merge trees); otherwise, ignore it and move on, The marked nodes constitute MST. a 9 11 i e h 1 g f 1 1 Figure 3. from CLRS KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April 016 / 5

3 Disjoint set data structure (see CLRS Ch. ) Kruskal What Collection of disjoint dynamic subsets over some set of elements; Here subsets are the trees within forest Operations make set(x) create a new singleton set containing x find set(x) identify the subset to which x belongs union(x, y) combine the two subsets to which x and y belong. Implementation Linked structure; see CLRS 1.3; O(mα(n)) for any m operations (α very slowly growing function almost constant ). Algorithm MST Kruskal(G, w): A empty set for each v in G.V do make set(v) sort edges in G.E in nondecreasing order by weight for each edge (u, v) in G.E in order do if find set (u) find set (v) then A A union {(u, v)} union(u, v) return A KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 Running time Prim s the basic idea n, m = num. nodes, edges Initialization (n make sets): O(n) Sorting edges: O(m log n) Loop (m iterations): m find sets m unions i.e. mα(n) in all Total: O(m log m) Grow (single) tree S from some starting point At each step: Consider the fringe of S i.e. edges with one node inside tree and one outside Choose the cheapest such (u, v) and incorporate it (and node v) into the tree. (Note this is a safe choice so we always expand the tree safely.) α(n) log(n) KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5

4 P s algorithm in action b c 7 d Let Q be a priority queue containing the nodes. Each node initially has key except for r which has key 0. a 11 h i g f e 3 Create Q as above. Create empty tree T. u Q.remove\ min() return T 3 Figure 3.5 from CLRS KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 decoded Invariant Create Q as above; Create empty tree T. u Q.remove\ min() return T Priority queue Q: Tree T : Q holds nodes not yet in embryonic MST non-q nodes constitute tree-so-far for non-q nodes, encodes parent-child rel. in MST for Q nodes, encodes cheapest edge linking node to a non-q node KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 Invariant: 1 T (Q) is a subset of an MST for each v Q, edge in T touching v is the lightest edge joining v to any node in Q key(v) is the weight of that edge KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5

5 Claim 3 Claim: Let Q = nodes not in Q; T (Q) = edges from T joining nodes in Q, then invariant holds at the beginning (and end) of each iteration. Prim T (Q) Generic MST A Proof of Claim 3 (Sketch) Create Q as above. Create empty tree T. u Q.remove\ min() Invariant true initially: T (Q) empty Each iteration preserves the invariant: Node u chosen by remove min Implicitly, (i) u leaves Q and joins Q and (ii) edge (u, parent T (u)) joins T (Q) Edge (u, parent T (u)) safe by choice of u For loop updates key(v) for neighbours of u to maintain loop invariant only edges touching u need be re-considered KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 Running Time Application 1 Create Q as above. Create empty tree T. u Q.remove\ min\ element() Creation of Q: O(n) No. of while loop iterations: n Priority queue ops remove min element O(log n) n O(n log n) replace key O(log n) (m) (O(m log n)) Contribution of for loop: For u = x, #iterations = #neighbours of x; Overall, #iterations = m. Total running time: O(m log n). Generic problem how to link together a collection of objects cheaply where link cost is proportional to distance between objects. Examples Oil-collection pipelines: objects = wells; links = pipelines between wells; (link cost prop. to length) Computer networks: objects = individual sites; links = dedicated lines; (link cost = monthly rental fee) KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5

6 Picture Model as graph weighted G = (V, E) V = well positions; E = all possible well-to-well pipelines; edge weight equals distance equals cost. MST gives set of pipelines (edges) that connect every well-head; have minimal total length. (More sophisticated approach might take consider Non-uniform pipeline costs; Links other than well-to-well links.) KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April 016 / 5 Application Let undirected G represent a computer network where edge weights denote reciprocal of link bandwidths. Let T be an MST for G. Let e(u, v) denote max. edge in unique path π in T from u to v. Path π has value weighte(u, v). Deleting e(u, v) partitions nodes into disjoint sets. The value of path π is its maximum edge weight. How to determine for each pair u, v the minimum value path joining those nodes? KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 Claim: Each edge in G from S to V S weighs at least weighte(u, v). Every path π from u to v contains one such edge and must have value KH (1/10/16) at least weighte(u, Lecture II: v). Minimium Spanning Tree Algorithms April 016 / 5

7 Application 3 Picture Cluster Analysis Partition experimental data into clusters; Cluster members more closely related to one another than non-cluster members. Application Plagiarism detection: Data programs submitted Clusters suspiciously similar submissions? Medical testing: Data patient measurements for various symptoms; Clusters groups of similarly affected patients. KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5 KH (1/10/16) Lecture II: Minimium Spanning Tree Algorithms April / 5