String Algorithms

1. anagram (a word or phrase made by transposing the letters of another word or phrase)

Question: Find all the anagrams in a dictionary
Solution: 1) Sign each word in the dictionary so that words in the same anagram class have the same signature, and then 2) bring together words with equal signatures. A signature scheme based on sorting is as follow: order the letters within the word alphabetically. To bring together, we can sort the words in the order of their signatures.

2. Palindrome ( a word, verse, or sentence (as *Able was I ere I saw Elba*) or a number (as 1881) that reads the same backward or forward)

3. String rotation without extra memory, used in text editor (move lines)

For example, abcdefg -> efgabcd. Step 1: Reverse the substring separately. abcdefg->dcbagfe Step 2: Reverse the whole string: dcbagfe->efgabcd

Trie or Prefix Tree

Sorting in Linear time

1. Counting Sort (input consists of integers in a small range)
2. Radix Sort
3. Bucket Sort (input is uniformly distributed)

Any comparison sort algorithm requires nlgn comparisons in the worse case.

Counting Sort (Running time: theta(n))

Counting sort assume that each of the n input elements is an integer in the range 0 to k, for some integer k. The basic idea of counting sort is to determine, for each input element x, the number of elements less than x. This information can be used to place element x directly into its position in the output array. We need to do a slight modification to handle the situation in which several elements have the same value. In practice, we usually use counting sort when we have k = O(n), the running time is theta(n).

Counting sort is stable: numbers with the same value appear in the output array in the same order as they do in the input array, which is very important since counting sort is often used as a subroutine in radix sort.

Counting sort requires two other arrays: the array B[1...n] holds the sorted output, and the array C[0...k] provides temporary working storage.

The algorithm itself is not complicated, watch out boundary errors when you try to implement it.

Radix Sort

Radix-Sort(A, d)
1. for i <-- 1 to d
2. do use a stable sort to sort array A on digit i

Radix sort can be used to sort records of information that are keyed by multiple fields.

Given n d-digit numbers in which each digit can take on up to k possible values, Radix-Sort correctly sorts these numbers in theta(d(n+k)) time. When d is constant and k = O(n), radix sort runs in linear time.

Bucket Sort

Bucket sort, or bin sort, is a sorting algorithm that works by partitioning an array into a finite number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting algorithm. Bucket sort is a generalization of pigeonhole sort.

WinCVS

http://www.devdaily.com/wincvs/HowToUseWinCVS/

http://ikon.as/wincvs-howto/#startingnew

TightVNC

An enhanced version of VNC (an abbreviation for Virtual Network Computing) is a great client/server software package allowing remote network access to graphical desktops.

TightVNC can be used to perform remote control and administration tasks in Windows, Unix and mixed network environments.

http://www.tightvnc.com/

Dead Lock

Network Flow

Terms:

• A Flow in graph G must satisify 1) capacity conditions. For each e belong to G, 0<=f(e)<=Ce. 2) conservation conditions. For each node v other than s and t, the flow value over all edges entering node v equals the sum of flow values over all edges leaving node v.
  
• The capacity of a cut, denoted as c(A, B), is the sum of the capacities of all edges out of A.
  

Properties:

• The residual graph Gf has at most twice as many edges as G.
  

The value of every flow is upper-bounded by the capacity of every cut. Let f be any s-t flow, and (A, B) any s-t cut, Then the value of flow v(f) <= c(A, B) , which is the capacity of a cut (A, B).

Max-Flow Min-Cut Theorem

In every flow network, there is a maximum flow f and a minimum cut (A, B) so that v(f) = c(A,B). There can be many minimum-capacity cuts in a graph G. All edges out of A are completed saturated with flow, while all edges into A are completely unused.

Ford-Fulkerson Algorithm: this algorithm will terminate when there is no s-t path in the residual graph. The algorithm is as follows.

Initially f(e)=0 for all e in G
While there is an s-t path in the residual graph Gf
...Let P be a simple s-t path in Gf
...f'=augment(f,P) // the running time for this helper function is O(m)
...Update f to f'
...Update the residual graph Gf to be Gf'
Endwhile
Return f

Helper function
augment(f,P)
..Let b = bottleneck(P,f)
..For each edge (u,v) in P
....If e=(u,v) is a forward edge then then
......increase f(e) in G by b
....Else ((u,v) is a backward edge, and let e=(v,u))
.....decrease f(e) in G by b
....Endif
..Endfor
..Return ()






Edmonds-Karp Algorithm