算法分析与设计AlgoDALectureNotesW3章节.pptVIP

Last Section An Example Task Problem analyzing Statement of the problem Development of a Model Design of the algorithm Correctness of the algorithm Implementation Analysis and complexity of the algorithm Statement of the problem 已知网络中任意两点间建立链路的花费， 需建立一个最小费用的通讯网络, 其中任意节点间可以相互通讯。 Model Let G=(V,E) be a connected, undirected graph; w(u,v) is the cost for connecting u,v∈V; find an acyclic subset T E and connects all vertices in V, such that is minimized Algorithm A To find a minimum-weight spanning tree T in a weighted, connected network G with N vertices and M edges. Step 0. [Initialize] Set T ← a network consisting of N vertices and no edges; set H ← G. Step 1. [Iterate] While T is not a connected network do through step 3 od; STOP. Step 2. [Pick a lightest edge] Let (U,V) be a lightest (cheapest) edge in H; if T + (U,V) has no cycles then [add (U,V) to T] set T ← T + (U,V) fi. Step 3. [Delete (U, V) from H] Set H ← H – (U, V). Does it work? There are several questions we should ask about this algorithm; 1. Does it always STOP? 2. When it STOPs, is T always a spanning tree of G? 3. Is T guaranteed to be a minimum-weight spanning tree? 4. Is it self-contained (or does it contain hidden, or implicit, sub-algorithms)? 5. Is it efficient? Question 45 Step 1 requires a sub-algorithm to determine if a network is connected, and step 2 requires a sub-algorithm to decide if a network has a cycle. These two steps can make the algorithm ineffective in that these sub-algorithms might be time-consuming. Algorithm B To find a minimum-weight spanning tree T in a weighted, connected network G with N vertices and M edges. Step 0. [Initialize] Label all vertices “unchosen”; set T ← a network with N vertices and no edges; choose an arbitrary vertex and label it “chosen”. Step 1. [Iterate] While there is an unchosen vertex do step 2 od; STOP. Step 2. [Pick a lightest edge] Let (U, V) be a lightest edge