Minimum Spanning Trees Outline: MST

Transcription

1 Minimm Spanning Trees Otline: MST Minimm Spanning Tree Generic MST Algorithm Krskal s Algorithm (Edge Based) Prim s Algorithm (Vertex Based)

2 Spanning Tree A spanning tree of G is a sbgraph which is tree (acyclic) and connect all the vertices in V. Spanning tree has only V - edges. 3 Inpt: Minimm Spanning Tree Undirected connected graph G = (V, E) and weight fnction w : E R, Otpt: A Minimm spanning tree T : tree that connects all the vertices and minimizes w( T) = w(, v) Greedy Algorithms Generic MST algorithm Krskal s algorithm Prim s algorithm (, v) T 4

3 Example: MST Example: MST 6 3

4 Generic MST Algorithm Or goal is to bild a spanning tree by adding one edge at a time to a set A in a greedy fashion. Basically, we jst need to somehow garantee orselves that at each step, the crrent set can be extended to an MST. Strategy: Grow the MST one edge at a time, ensring that the partial soltion remains a sbset of some MST How do we do that? 7 Generic MST Algorithm Let s assme that or crrent set of edges A already satisfies the property that A can be extended to an MST. Qestion: What edges can we add to A to maintain the property? Answer: an edge e sch that w(e) w(other edges) and A {e} is acyclic We will call sch edge a safe edge if it also doesn t create a cycle when added to A. 4

5 Generic MST Algorithm Generic-MST(G, w). A = { }. while A does not form a spanning tree 3. find an edge (, v) that is safe for A 4. Add (, v) to A. retrn A How to find a safe edge? Definitions: Generic MST Algorithm A ct (S, V - S) of G = (V, E) is a partition of V into sets An edge (, v) E crosses the ct (S, V - S) if one point is in S while the other point in V - S A ct respects a set A of edges if no edges in A crosses the ct. An edge is light if its weight is minimm of all edges crossing the ct

6 Generic MST Algorithm S V - S a 4 b 7 c d i h g f e S V - S black vertices S = (a, b, d, e) ct white vertices V-S = (c, i, h, g, f) red line ct ble edges crossing the ct (c, d) only light edge green a sbset A of edges; which is a ct (S, V-S) respects to A, since no edge of A crosses the ct. Theorem 3. (CLRS) Let G = (V, E) be a connected, ndirected graph w is weight fnction w : E R Let A E be inclded in some MST T for G Let (S, V - S) be any ct of G that respect A Let (, v) be a light edge (min-weight) crossing the ct (S, V - S) Then, edge (, v) is safe for A i.e., (,v) T, a MST of G 6

7 Proof sppose (, v) T Theorem 3. (CLRS) look at path from to v in T swap (x, y) with (, v) the first edge on path from to v in T that crosses from S to V S this increases w(t) contradiction T spposed to be MST S x y v V-S 3 Generic MST Algorithm Corollary 3. (CLRS): Let A E be inclded in some MST Let C = (V C, E C ) and C = (V C, E C ) be two distinct connected components (trees) in the forest G A = (V, A). If (, v) is a light edge crossing the ct = (V C, V C ) then (, v) is safe for A. 7

8 Generic MST Algorithm Generic-MST(G, w). A = { }. while A does not form a spanning tree 3. find an edge (, v) that is safe for A 4. Add (, v) to A. retrn A As the algorithm proceeds: A is always acyclic At any point of the exection of the algorithm, Graph G A = (V, A) is a forest, and Each connected component is a tree Also, any safe edge (, v) for A Connects distinct components of G A, Since A (, v) mst be acyclic Generic MST Algorithm Generic-MST(G, w). A = { }. while A does not form a spanning tree 3. find an edge (, v) that is safe for A 4. Add (, v) to A. retrn A The loop in lines -4 is exected V - times Initially when A = {}, there are V trees in G A Each tree has only one vertex When G A (forest) contains only a single tree, the algorithm terminates 6

9 Edge based algorithm Greedy strategy: Krskal's Algorithm From the remaining edges, select a least-cost edge that does not reslt in a cycle when added to the set of already selected edges Repeat V - times 7 INPUT: Krskal's Algorithm edge-weighted graph G = (V, E), with V = n OUTPUT: a spanning tree A of G toches all vertices, has n- edges of minimm cost ( = total edge weight) Algorithm: Start with A empty, Add the edges one at a time, in increasing weight order An edge is accepted, if it connects vertices of distinct trees (if the edge does not form a cycle in A) ntil A contains n- edges

10 Krskal's Algorithm MST-Krskal(G,w) A for each vertex v V[G] do 3 Make-Set(v) 4 sort the edges of E by nondecreasing weight w for each (,v) E, in nondecreasing of weight do 6 if Find-Set() Find-Set(v) then 7 A A {(,v)} Union(Set(),Set(v)) retrn A Krskal's Algorithm Lines -3 initialize the set A to empty set and create V trees, one containing each vertex. The edges in E are sorted into nondecreasing order by weight in line 4. The for loop in lines - checks, for each (, v), whether the endpoints and v belong to the same tree. If they do, then the edge (, v) cannot be added to the forest withot creating a cycle, and the edge is discarded. Otherwise, the two vertices belong to different trees. In this case, the edge (, v) is added to A in line 7, and the vertices in the two trees are merged in line. 0

11 Data Strctres For Krskal s Algorithm Does the addition of an edge (, v) to A reslt in a cycle? Each component of A is a tree. When and v are in the same component, the addition of the edge (, v) creates a cycle different components, the addition of the edge (, v) does not create a cycle Data Strctres For Krskal s Algorithm Each component of A is defined by the vertices in the component. Represent each component as a set of vertices. {,, 3, 4}, {, 6}, {7, } Two vertices are in the same component iff they are in the same set of vertices

12 Data Strctres For Krskal s Algorithm When an edge (, v) is added to A, the two components that have vertices and v combine to become a single component In or set representation of components, the set that has vertex and the set that has vertex v are nited. {,, 3, 4} + {, 6}{,, 3, 4,, 6} Data Strctres For Krskal s Algorithm Initially, A is empty Initial sets are: {} {} {3} {4} {} {6} {7} {} Does the addition of an edge (, v) to A reslt in a cycle? If not, add edge to A s = Find-Set(); s = Find-Set(v); if (s s ) then Union(s, s ); 4

13 Krskal s Algorithm ? 7? Krskal s Algorithm 7 7? 3 3 7? ?

14 Krskal s Algorithm ? 7? ? Krskal s Algorithm? 7 7 3? 3?

15 Krskal's Algorithm MST-Krskal(G,w) A for each vertex v V[G] do 3 Make-Set(v) 4 sort the edges of E by nondecreasing weight w for each (,v) E, in nondecreasing of weight do 6 if Find-Set() Find-Set(v) then 7 A A {(,v)} Union(Set(),Set(v)) retrn A Rnning Time of Krskal s Algorithm Krskal s Algorithm consists of two stages. Initializing the set A in line takes O() time. Sorting the edges by weight in line 4. takes O(E lg E) Performing V MakeSet() operations for loop in lines -3. E FindSet(), for loop in lines -. V - Union(), for loop in lines -. which takes O(V + E) The total rnning time is O(E lg E) Observing that E < V we have lg E = O(lg V), So total rnning time becomes O(E lg V). 30

16 Prim s Algorithm Prim s algorithm has the property that edges in the set A always form a single tree. The tree starts from an arbitrary root vertex r and grows ntil the tree spans all the vertices in V. At each step, a light edge is added to the tree A that connects A to an isolated vertex of G A = (V, A). Adds only edges that are safe for A. When algorithm terminates, edges in A form MST. MST A for G: A={(v, p[v]): vεv-{r}}. Vertex based algorithm. Grows one tree, one vertex at a time 3 Prim s Algorithm MST-Prim(G,w,r) //G: graph with weight w and a root vertex r for each V[G] key[] 3 p[] NIL 4 key[r] 0 Q = BildMinHeap(V,key); // Q vertices ot of T 6 while Q do 7 ExtractMin(Q) // making part of T for each v Adj[] do if v Q and w(,v) < key[v] then p[v] key[v] w(,v) 3 pdating keys All vertices that are not in tree reside in min-priority qee Q based on a key field. For each vertex v, key[v] is min weight of any edge connecting v to a vertex in tree. key[v]= if there is no edge and p[v] names parent of v in tree. When algorithm terminates the min-priority qee Q is empty. 6

17 Prim s Algorithm Lines - set the key of each vertex to (except root r, whose key is set to 0 first vertex processed). Also, set parent of each vertex to NIL, and initialize min-priority qee Q to contain all vertices. Line 7 identifies a vertex єqincident on a light edge crossing ct (V-Q, Q) (except first iteration, =r de to line 4). Removing from set Q adds it to set Q-V of vertices in tree, ths adding (, p[]) to A. The for loop of lines - pdate key and p fields of every vertex v adjacent to bt not in tree. 33 Rn on example graph

18 Rn on example graph key[] = 3 Rn on example graph 6 4 r 0 3 Pick a start vertex r 36

19 Rn on example graph Red vertices have been removed from Q 37 Rn on example graph Red arrows indicate parent pointers 3

20 Rn on example graph Rn on example graph

21 Rn on example graph Rn on example graph

22 Rn on example graph Rn on example graph

23 Rn on example graph Rn on example graph

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27 Rn on example graph Prim s Rnning Time What is the hidden cost in this code? Extract-Min is exected V times MST-Prim(G,w,r) for each V[Q] Decrease-Key is exected O( E ) times key[] 3 p[] NIL while loop is exected V times 4 key[r] 0 Q = BildHeap(V,key); //Q vertices ot of T 6 while Q do 7 ExtractMin(Q) // making part of T for each v Adj[] do if v Q and w(,v) < key[v] then p[v] key[v] w(,v) DecreaseKey(v, w(,v)); pdating keys 4 7

28 Prim s Rnning Time Time complexity depends on data strctre Q Binary heap: O(E lg V): BildHeap takes O(V) time nmber of while iterations: V ExtractMin takes O(lg V) time total nmber of for iterations: E DecreaseKey takes O(lg V) time Hence, Time = V + V.T(ExtractMin) + E.T(DecreaseKey) Time = O(V lg V + E lg V) = O(E lg V) Since E V (becase G is connected)