1. Greedy
Minimum Spanning Tree: Prim's Algorithm
研究⽣生：鍾聖彥
教授： 許慶昇 ⽼老師

2. Key ingredients of greedy
algorithms
• Optimal substructure: An optimal solution to the
problem contains within it optimal solutions to
subproblems.
• Greedy-choice property: We can assemble a
globally optimal solution by making locally optimal
(greedy) choices.

3. 什麼問題可以使⽤用greedy
algorithm來解?
1.如果做出⼀一個choice之後,
可以找到剩下要解的單⼀一個
subproblem
2.Greedy-choice property(必須證明最佳解裡⾯面⼀一定有
greedy choice)
3.有optimal substructure (必須證明⼤大問題的最佳解裡
⾯面有⼩小問題的最佳解)

4. Graphs

5. EXAMPLE：Weighted graph
Ａ
Ｃ
Ｄ
Ｆ
Ｅ
Ｂ
Ｇ
Ｈ

6. Spanning Tree(⽣生成樹)

7. Minimum Spanning Tree
Ａ
Ｃ
Ｄ
Ｆ
Ｅ
Ｂ
Ｇ
Ｈ
⺫⽬目的：移除weighted graph中的部份邊線，使得⼦子圖仍然
保持連通性，且邊線weight總和為最⼩小
權重最⼩小的⽣生成樹就是最⼩小⽣生成樹

8. Application

9. Optimal substructure
• ⼀一、兩棵 MST ，要合併成⼀一棵 MST 時，以兩棵 MST 之間權重最⼩小
的邊進⾏行連結，當然會是最好的。
• ⼆二、三棵 MST ，要合併成⼀一棵 MST 時，先連結其中兩棵連結權重
最⼩小的 MST ，然後才連結第三棵，總是⽐比較好。
• 三、⼀一個單獨的點，可以視作⼀一棵 MST 。
• 由以上三點，可以歸納出⼀一個 greedy 演算法：以權重最⼩小的邊連結
各棵 MST ，⼀一定⽐比較好。

10. Optimal substructure
MST T

11. Optimal substructure
MST T
Remove any edge (u,v) ∈ T

12. Optimal substructure
MST T
Remove any edge (u,v) ∈ T. Then it is partitioned into two
subtrees T1 and T2
T1
T2

13. Optimal substructure
MST T
Theorem. The subtree T1 is MST of G1 = (V1,E1),the subgragh
of G induced by the vertex of T1
Similarly for T2
T1
T2
G1

14. Proof of Optimal substructure
MST T
T1
T2
G1
w(T) = w(u, v) + w(T1) + w(T2)
If T1′ were a lower-weight spanning treat than T1
for G1, then T′ = {(u, v)} ∪ T1′ ∪ T2 would be a
lower-weight spanning tree than T of G

15. –Greedy Agorithm
Greedy-choice property"
A locally optimal choice is globally optimal

16. Theorem
Let T be the MST of G = (V,E), and let A ⊆ V.
Suppose that (u,v)∈ E is the least-weight edge
connecting A to V-A, then (u,v)∈ T
The proof is by contradiction(⽭矛盾)
Cut-and-Paste is a way used in proofing graph theory concepts, Idea: Assume you have
solution for Problem A, you want to say some edge/node, should be available in solution.
You will assume you have solution without specified edge/node, you try to reconstruct a
solution by cutting an edge/node and pasting specified edge/node and say new solution
benefit is at least as same as previous solution.

17. Proof of theorem
Consider the unique simple path from u to v in T

18. Proof of theorem
Swap(u, v) with the first edge on this path that
connects a vertex in A to a vertex in V–A

19. Proof of theorem
A lighter-weight spanning tree than T results.
Proving that greedy choice is good enough.

20. Prim’s Algorithm

21. Prim’s Algorithm
Use Priority Queue

22. Prim’s Algorithm
• 增加節點的觀念做為出發點。
• ⾸首先以某⼀一節點當作出發點，在與其相連且
尚未被選取的節點裡，選擇權重最⼩小的邊，
將新的節點加⼊入。
• 如此重覆加⼊入新節點，直到增加了n - 1條邊
為⽌止。(假設有 n 個節點)