ADVANCED ALGORITHMS Lecture 11: Review, flows/cuts ANNOUNCEMENTS - - PDF document

ADVANCED ALGORITHMS

Lecture 11: Review, flows/cuts

ANNOUNCEMENTS

➤ HW 1 grades out (?) ➤ HW 2 due this Friday, 11:59 PM

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TODAY’S PLAN

➤ Quick review/recap of course so far ➤ Comments about HW ➤ Flows and cuts in graphs ➤ Some applications of flows

DIVIDE & CONQUER

➤ Basic idea: divide instance into sub-problems, solve them recursively,

combine

➤ Usually sub-problems don’t overlap much ➤ Analysis by writing down a recurrence ➤ Usually prove correctness via induction ➤ How to solve a recurrence? (plug-n-chug, guess and prove, recurrence

tree, Akra-Bazzi, master theorem, …)

➤ Examples: multiplying n-bit numbers, merge sort, median finding

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n time

DYNAMIC PROGRAMMING

➤ Basic idea: write down a recursive expression for quantity of interest,

store answers to all “sub-problems”

➤ Works well if # of distinct sub-problems is small ➤ Examples: subset sum, coin change, longest increasing subsequence,

…

➤ Key: how we write recurrence determines space/time ➤ See notes: top down vs bottom up

EXAMPLE — COIN CHANGE

➤ Problem: given k denominations of coins d1, d2, …, dk, find how to

make change for S cents using fewest number of coins

➤ One way: let m(S, i) be the min number of coins needed to make

change for S using coins di and above. Recurrence tries all possibilities for # of di coins:

m(S, i) = min

j : jdi≤S j + m(S − jdi, i + 1)

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EXAMPLE — COIN CHANGE

➤ Problem: given k denominations of coins d1, d2, …, dk, find how to

make change for S cents using fewest number of coins

➤ One way: let m(S, i) be the min number of coins needed to make

change for S using coins di and above. Recurrence tries all possibilities for # of di coins:

➤ Another: let f(S) be the min number of coins needed to make change

for S. Recurrence tries all possible coins:

m(S, i) = min

j : jdi≤S j + m(S − jdi, i + 1)

f(S) = min

i : di≤S f(S − di)

f s

min

• f coins needed tomake

changefor S

Observation

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DYNAMIC PROGRAMMING

➤ Basic idea: write down a recursive expression for quantity of interest,

store answers to all “sub-problems”

➤ Works well if # of distinct sub-problems is small ➤ Examples: subset sum, coin change, longest increasing subsequence,

…

➤ Key: how we write recurrence determines space/time ➤ See notes: top down vs bottom up

GREEDY ALGORITHMS

➤ Basic idea: to make a sequence of choices, make one by one, best

choice in each step. Greedy == myopic

➤ Typically: ➤ easy to design algorithm & implement ➤ Not easy to analyze ➤ Doesn’t produce optimal answers ➤ Examples: matching gifts to children, scheduling, spanning trees,

hiring problem — first example of approximation — always at least 63% of optimum!

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value

LOCAL SEARCH

➤ Basic idea: start with an arbitrary solution, and try to improve ➤ Typically: ➤ easy to design algorithm & implement ➤ how we swap determines running time (usually large) ➤ produces reasonable answers (esp in practice) ➤ Examples: matching gifts to children (always 1/2 of optimum),

gradient descent is example of local search

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BASIC GRAPH ALGORITHMS

➤ Shortest paths ➤ Problem: given a directed graph with edge lengths, find the shortest

total length path from u to v. Can be done efficiently

➤ Bellman-Ford/Shimbel’s algorithm: maintain “candidate” dist[w]

values for all vertices w. Update all using 
 dist[w] = min(dist[w], dist[i] + len(i, w)) for all edges (i,w)

➤ Takes n iterations, each taking m = |E| time ➤ Cool property: works also when graph has negative edges (but no

cycles)

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BASIC GRAPH ALGORITHMS

➤ Shortest paths ➤ Problem: given a directed graph with edge lengths, find the shortest

total length path from u to v. Can be done efficiently

➤ Dijkstra’s algorithm: maintain “candidate” dist[w] values for all

vertices w. Grow “ball of certain radius” around u.

➤ Add one vertex to ball in each iteration — pick one that has least

dist[] value; update dist[] values

➤ Properties: near linear time; doesn’t work with negative edges

Very similar to

minimum spanning hie

NEXT: CUTS & FLOWS

CUTS IN GRAPHS

➤ Problem: given a graph, find a small set of (weighted) edges to “cut”

so that the graph is disconnected.

➤ Variants: ➤ Separate given vertices u and v (called s-t cut) ➤ Break graph into two nearly equal pieces (partition) ➤ Maximize the number of edges going between the two pieces ➤ Each variant has many applications

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MIN S-T CUT

➤ Problem: given a directed graph with edge weights, find a small set of

(weighted) edges to “cut” so that there is no longer a direct path from s to t.

➤ origins

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History

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Radndomized algorithm 2015

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MAXIMUM FLOW

Problemi

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a weighted directed graph

intea wt of

an edge is

interpreted as the

capacity

Find

the

maximum

amount of

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source

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to a t

sink

that does not violate

the

edge capacities 50

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ALGORITHMS FOR FLOW?

Natural approaches

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capacities appropriately

find if

another path exists etc does Not work

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MAXIMUM FLOW VS MIN CUT

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