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Unformatted text preview: Announcements:
MP7 due 5/4, 11 :59p.
Final exam: 5/11, 710p, rooms on web.
Review sessions: 5/4. _7_1_gp. 1404 Siebel
_5_/_8_. 1404 Siebel
Today:
Graphs — Minimum Spanning Trees
Single Source Shortest Palh How do we get from here to there? Need: I. pom/non 'v/or (Via/(21V
2. éi(gp/r I.01/9/PMPI7f{Zf;OI7
3. '/?(zvm.sa/.D/* 5, q. rf/Bnrif/IMS. (23 Mfg'f' A) Sham5r Pam Minimum Spanning Tree Algorithms: Input: connected. undirected graph G with unconstrained edge weights
Output: a graph G' with the following characteristics  06‘ is a spanning subgraph of G 06‘ is connected and acyclic (a tree) 06' has minimal total weight among all such spanning trees I155 Example of Prim‘s algorithm  initialize structure:
1. For al v.d[v] = “inﬂnity'. plv] = rLulI ) ‘ ~
Initialize source: dls] = 0 Initialize priority (mh) queue Initialize set of .beled v  .. to (2). Repeat these steps mimes: 1. Find & remove minimum d unlabeled
vertex: v H. [3“ A: r
2. Label vertex v 3.'For al unlabelled neiggbors w o . if cost(v._w) < dlg]
dlw] = oost(v.w) I":
piw1= v r ‘4 D Prim’s Algorithm (undirected graph with unconstrained edge weight Initialize structure:
1. For 3! v. d[v] = "inﬁnity". piv] = nul
Initialize source: dls] = 0 2.
3. Initialize priority (min) queue
4. initialize set of labeled vertices to (2}. Repeat these steps Qtimes:
1. Find minim md nla erte :  
u .03 W " " WhICh IS best? .bﬁp'nJS m density or“ (/1? shy/l fémar5r ‘.' For el unlabelled neighbors w of v. n 3H6? t  2. Label vertex v uqiq Single source shortest path Given a start vertex (scurce) 3. find the path of
least total cod from s to every vertex in the
graph. Single source shortest path: Input: directed graph G with nonnegative edge weights. and a start vertex 3.
’ Output: A subgraph Q' consisting of the shortest (minimum total cost)
paths from s to every other vertex in the graph. " ' Dijkstra's Algorithm (1959) Single source shortest path (directed graph w nonnegative edge weights):
Dijkstra‘s Algorithm (1959) Given a source vertex 3. we
wish to ﬁnd the shortest path
from s to every other vertex
in the graph. Initialize structure: Repeat these steps: 1. Label 8 new (unlabelled) vertex v.
whose shortest distance has been found 2. Update v's neighbors with an
improved distance Single source shortest path a pretty good applet... http://www.de.toronto.odulpeoplolJamosStmn/270I979881Laffraloiksu'aApplet.hmi Single source shortest path (directed graph w nonnegative edge weights): Initialize structure: 1. For all v. d[v] = "inﬁnity". p{v] = nul
2. Initialize source: dls] = 0 3. Initialize priority (min) queue Repeat these steps n umes:
1. Find minimum d unlabeled vertex: v 2. Label vertex v 3. For all unlabeiled neighbors w of v. “@126” :zijj‘v’wmm Running time? Single source shortest path (directed graph w nonnegative edge weights): initialize structure: ed] list 1. For al v. d[v] = "inﬂnity". p[v] = nul 2. initialize source: dls] = 0 0i" '09 n t m I09 0)
3. Initialize priority (min) queue 4. Initialize set of labeled vertioes to 0.
CW) Repeat these steps n times: 1 . Find minimum d unlabeled vertex: v 2. Label vertex v S
3. For 8! unlabelled neighbors w of v. i
if oost(v.w) < dlw] P'f i“ .5 dlw] = dlv] + oost(v.w) plwl=v How do we get from here to there? Need: I. pom/non 'v/or (Via/(21V 2. éi(gp/r [.07/9/PMP’7fflfl.O’7 3. '/?(zvm.sa/.D/* 5, A]. fz/yﬁl‘l.'(‘/IM.5. '
(23 MST Kruslcal‘s. Rn»)
A) Sham5r Pam Dowa's ...
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 Spring '09
 Heeren
 Graph Theory, Siebel, source shortest path

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