2. • Optimal structure: optimal solution to problem
• consists of optimal solutions to subproblems
• Overlapping subproblems: few subproblems in total,
• many recurring instances of each
• Solve in “near linear” time through recursion, using a
table to show state of problem at any given moment of
time
• Reduces computation time from exponential to linear in
some cases
Dynamic programming
3. Greedy algorithm always makes the
choice that is the locally optimal (best) at
the moment
It can reduce complex problem to linearly
solvable time, and can be much easier to
code
3
Greedy methods
4. Shortest-path problem and multiple instance
knapsack problems can be solved using
greedy approach
Activity selection is another example
4
Greedy methods
5. Problem: Stampede midway problem
» Buy a wristband that lets you onto any ride
» Lots of rides, each starting and ending at different times
» Your goal: ride as many rides as possible
• Alternative goal that we don’t solve here: maximize
time spent on rides
5
Greedy methods
6. Let A be an optimal solution of S and let k be the
minimum activity in A (i.e., the one with the
earliest finish time). Then A - {k} is an optimal
solution to S’ = {i ∈ S: si ≥ fk}
» In words: once activity #1 is selected, the problem
reduces to finding an optimal solution for activity
selection over activities in S compatible with #1
6
Greedy algorithm
7. The algorithm is following:
» Sort the activities by finish time
» Schedule the first activity
» Then schedule the next activity in sorted list which
starts after previous activity finishes
» Repeat until no more activities
Idea:
Always pick the shortest ride available at the time
Greedy-choice Property.
» A globally optimal solution can be arrived at by
making a locally optimal (greedy) choice.
7
Greedy Algorithm
8. Backtracking can reduce a NP compete
problem to linear problem by only going
through selected branches of the global
solution.
8
Backtracking
9. Graph-coloring problem
8 queens problem
4 knight problem
Perfect hash function problems
are examples of backtracking method
9
Backtracking
10. The goal is to color vertices in a graph G={V,E} so that no 2 adjacent
vertices have the same color. Partial 3-coloring problem means only 3
colors are considered.
Direct approach builds the tree of ALL possibilities in exponential time.
10
Backtracking
11. Partial 3-coloring (3 colors) is solved by the following method:
Color first vertex with 1st color, color next vertex with the next color,
check if those two vertices are adjacent, if not - coloring is legal,
proceed to next vertex, if yes and color is the same – coloring is illegal,
try next color for second vertex. If all colors tried and all colorings are
illegal, backtrack, try next color for previous vertex etc.
Note: sometimes solution is impossible.
Exponential O(3^n) complexity is reduced to O(n) on average.
11
Backtracking
12. Finally, job scheduling problem can be solved
by converting the algebraic relationship xi-
xj>=c to directed graph with vertices xi, xj,
direction from xj to xi, and edge cost (or time
required to complete job xj) is c.
The problem of finding time (minimum) when
the last activity can commence (i.e. all
preceding activities has been completed) is
then converted to longest path problem on
the graph.
12
Job scheduling problem
