# Material for the algorithms class taught by Emanuele "Manu" Viola.

## Description

Presents the mathematical techniques used for the design and analysis of computer algorithms. Focuses on algorithmic design paradigms and techniques for analyzing the correctness, time, and space complexity of algorithms. Topics may include asymptotic notation, recurrences, sorting and searching, divide and conquer, data structures, hashing, greedy algorithms, dynamic programming, graph algorithms, and NP-hardness.

We will be following the video lectures below. The video lectures are in turn based on Manu's slides. However, we list next a few books that cover overlapping material:
• [E] Algorithms, by J. Erickson
• [CLRS] Introduction to Algorithms, third edition, by T. Cormen, C. Leiserson, R. Rivest, and C. Stein
• [DPV] Algorithms, by S. Dasgupta, C.H. Papadimitriou, and U.V. Vazirani
• [KT] Algorithm Design, by Jon Kleinberg, Éva Tardos
Some of the material we cover will be written down here:
Note: this text is in a preliminary stage, but it may be useful as an extra reference.

# Video lectures

### Video 1

• Introduction to algorithms. Defines algorithm.
• Extra pointers: [E, 0] [CLRS, 1]

### Video 2

• Measuring the performance of algorithms. Asymptotic notation Big-Oh, Omega, and Theta. Length: 27:05 (note: this is longer than usual. A good breaking point is right before we introduce Big-Omega)
• Extra pointers: [CLRS, 3]

### Video 3

• Bubble sort.
• Extra pointers: [CLRS, Problem 2-2]

### Video 4

• Counting sort.
• Extra pointers: [CLRS, 8.2]

### Video 5

• Extra pointers: [CLRS, 8.3]

### Video 6

• Divide and conquer paradigm, mergesort.
• Extra pointers: Mergesort [E, 1.4], [CLRS, 2.3]

### Video 7

• Quicksort.
• Extra pointers: Quicksort [E, 1.5], [CLRS, 7]

### Video 8

• Oblivious mergesort
• Extra pointers: [V, 2.1]

### Video 9

• Review of sorting. Computing the median [E, 1.7], [CLRS, 9.3]

### Video 10

• Closest pair [CLRS, 33.4]

### Video 11

• Karatsuba's integer multiplication [E, 1.8], [KT 5.5]

### Video 12

• Strassen's matrix multiplication [CLRS, 4.2], Fast Walsh-Hadamard transform [V, 2.2]

### Video 13

• Dynamic programming, coin change

### Video 14

• Longest common subsequence [CLRS, 15.4]

### Video 15

• Dynamic programming in economics [V, 3.1]

### Video 16

• Subset sum [E, 3.8]

### Video 17

• Greedy algorithms, activity selection problem [E, 4.2], [CLRS, 16.1]

### Video 18

• Data structures, binary search trees [CLRS 12.1, 12.2] [V 4]

### Video 19

• Rotations, AA trees [CLRS 13.2] [V 4]

### Video 20

• Insertion and deletion in AA trees [V 4]

### Video 21

• Hash functions [CLRS 11]

### Video 22

• Queues and heaps [CLRS 6]

### Video 23

• Graphs, breadth-first-search [E 5.4], [CLRS 22.1], [E 8.4], [CLRS 22.2]

### Video 24

• Bellman-Ford weighted minimum distance [E 8.7] [CLRS 24.1]

### Video 25

• Dijkstra shortest path [E 8.6], [CLRS 24.3]

### Video 26

• All pairs shortest path. All-pairs shortest-path via matrix multiplication [E 9.7], [CLRS 25.1]. Floyd Warshall [E 9.8], [CLRS 25.2]. Johnson's algorithm [E 9.4], [CLRS 25.3]

### Video 27

• Flow networks [E 10.1 - 10.3] [CLRS 26.1] (our notation is closer to edition 2 CLRS). Ford-Fulkerson [E 10.4] [CLRS 26.2]

### Video 28

• Edmonds-Karp [E 10.6] [CLRS 26.3]. Maximum matching [E 11.3], [CLRS 26.4].

### Video 29

• 3SAT reduces to CLIQUE [CLRS 34.5.1]

### Video 30

• CLIQUE to VERTEX-COVER [CLRS 34.5.2]

### Video 31

• 3SAT reduces to SUBSET-SUM [CLRS 34.5.5]

### Video 32

• 3SAT reduces to 3COLOR [E 12.10]

## Known bugs in videos and slides

NOT FIXED NEITHER IN SLIDES NOR VIDEOS

odd-even-merge: for i should go to n-3, not n-1.

The h-k-1 in the slide about selecting the h-th smallest element should be h-k.

DP & Video 16 at time 7:05 n = 3 should be n = 5

In the activity selection pseudocode, when we add we should add a[m], not a[i].

In the heap lecture video (Video 22):
Starting from 13:45, 10 should be 1.

There is a typo on the left-hand graph on the flow slides 16 and 17. The 30/30 in the middle edge should be 0/30.

FIXED IN divide-and-conquer-annotated BUT NOT on videos and not on non-annotated version:

Mergesort base case is high = low, not high-low < 1
Same for quicksort
Partition: Do j-- while A[j] > A[hi] -> Do j-- while A[j] > A[hi] & i < j
Running time bound for selecting h-th smallest element: inequalities are on the wrong side.

## Tentative list of topics to be covered.

Asymptotic notation. [CLRS, 3]
Bubblesort [CLRS, Problem 2-2]
Countingsort [CLRS, 8.2]
Divide and conquer:
Mergesort [E, 1.4], [CLRS, 2.3]
Quicksort [E, 1.5], [CLRS, 7]
Oblivious mergesort [V, 2.1]
Median [E, 1.8], [CLRS, 9.3]
Closest pair [CLRS, 33.4]
Karatsuba's integer multiplication [E, 1.9], [KT 5.5]
Strassen's matrix multiplication [CLRS, 4.2]
Polynomials and fast Fourier transform [CLRS, 30], [KT, 5.6]
Dynamic programming:
Coin-change problem
Longest common subsequence [CLRS, 15.4]
Subset sum [E, 3.8]
Dynamic programming in economics [V, 3.1]
Greedy:
Activity selection [E, 4.2], [CLRS, 16.1]
Data structures:
Binary search trees [CLRS 12.1, 12.2] [V 4]
Rotations [CLRS 13.2] [V 4]
AA Trees [V 4]
Hash functions [CLRS 11]
Heaps [CLRS 6]
Graph algorithms:
Graph representations [E 5.4], [CLRS 22.1]
Breadth-first search [E 8.4], [CLRS 22.2]
Bellman-Ford [E 8.7] [CLRS 24.1]
Dijkstra [E 8.6], [CLRS 24.3]
All-pairs shortest-path via matrix multiplication [E 9.7], [CLRS 25.1]
Floyd Warshall [E 9.8], [CLRS 25.2]
Johnson's algorithm [E 9.4], [CLRS 25.3]
NP completeness:
3SAT reduces to CLIQUE [CLRS 34.5.1]
CLIQUE to VERTEX-COVER [CLRS 34.5.2]
3SAT reduces to SUBSET-SUM [CLRS 34.5.5]
3SAT reduces to 3COLOR [E 12.10]
Maximum flow:
Flow networks [E 10.1 - 10.3] [CLRS 26.1] (our notation is closer to edition 2 CLRS)
Ford-Fulkerson [E 10.4] [CLRS 26.2]
Edmonds-Karp [E 10.6] [CLRS 26.3]
Maximum bipartite matching [E 11.3], [CLRS 26.4]
Linear programming:
The linear programming problem [CLRS 29.1, 29.2]
The simplex algorithm [CLRS 29.3]
Duality [CLRS 29.4]
Approximation algorithms:
2-approximation for VERTEX-COVER [CLRS 35.1]
2-approximation for weighted VERTEX-COVER [KT 11.6]
Vector programs, randomized rounding, and max-cut