AlgorithmsDefinition | unambiguous procedure executed in a finite number of steps | What makes a good algorithm? | Correctness, Speed, Space, Simplicity | Speed: | time it takes to solve problem | Space: | amount of memory required | Simplicity: | easy to understand, easy to implement, easy to debug, modify, update |
Running TimeDefinition | measurement of the speed of an algorithm | Dependent variables: | size of input & content of input | Best Case: | time on the easiest input of fixed size | meaningless | Average Case: | time on average input | good measure, hard to calculate | Worst Case: | time on most difficult input | good for safety critical systems, easy to estimate |
Proofs by Induction (Examples)
Loop InvariantsDefinition | loop property that holds before and after every iteration of a loop. | Steps: | 1. Initialization | If it is true prior to the iteration of the loop | 2. Maintenance | If it is true before an iteration of the loop, it remains true before the next iteration | 3. Termination | When the loop terminates, the invariant gives us a useful property that helps show the algorithm is correct |
QuickSortDivide: | choose an element of the array for pivot | | | divide into 3 sub-groups; those smaller, those larger and those equal to pivot | Conquer | recursively sort each group | Combine | concatenate the 3 lists |
QuickSortAlgorithm partition(A, start, stop)
Input: An array A, indices start and stop.
Output: Returns an index j and rearranges the elements of A
such that for all i<j, A[i] ! A[j] and
for all k>i, A[k] " A[j].
pivot # A[stop]
left # start
right # stop - 1
while left ! right do
while left ! right and A[left] ! pivot) do left # left + 1
while (left ! right and A[right] " pivot) do right # right -1
if (left < right ) then exchange A[left] $ A[right]
exchange A[stop] $ A[left]
return left |
Time Complexities:
• Worse case:
– Already sorted array (either increasing or decreasing)
– T(n) = T(n-1) + c n + d
– T(n) is O(n2)
• Average case: If the array is in random order, the
pivot splits the array in roughly equal parts, so the
average running time is O(n log n)
• Advantage over mergeSort:
– constant hidden in O(n log n) are smaller for quickSort.
Thus it is faster by a constant factor
– QuickSort is easy to do “in-place” In Place SortingDefinition: | Uses only a constant amount of memory in addition of that used to store the input | Importance: | Great for large data sets that take up large amounts of memory | Examples: | Selection Sort, Insertion Sort (Only moving elements around the array) | MergeSort: | Not in place: new array required |
| | Object Orientated ProgrammingDefinition: | User defined types to complement primitive types | | | Called a class | Contains: | Data & methods | Static members: | members shared by all objects of the class |
Recursion ProgrammingDefinition | using methods that call themselves | Structure: | base case | a simple occurrence that can be answered directly | recursive case | A more complex occurrence of the problem that cannot be directly answered, but can instead be described in terms of smaller occurrences of the same problem. |
Divide & ConquerDivide | the problem into sub problems that are similar to the original but smaller in size | Conquer | the sub-problems by solving them recursively. If they are small enough, solve them in a straightforward manner | Combine | the solutions to create a solution to the original problem |
BIG O Definitionf(n) & g(n) are two non negative functions defined on the natural numbers N | f(n) is O(g(n)) if and only if: | there exists an integer n0 and a real number c such that Ɐn>= n0 f(n) <= c * g(n) | | | N.B. The constant c must not depend on n |
Big O RecurrenceSum of ai from 0 to n = (a(n+1) -1 )/(a-1) Limitations of ArraysSize has to be known in advance | memory required may be larger than number of elements | inserting or deleting an element takes up to O(n) |
ADT: QueuesFIFO(First in first out) | Any first come first serve service | Operations | enqueue() - add to rear | | | dequeue() - removes object at front | | | front() - returns object at front | | | size() - returns number of objects O(n) | | | isEmpty() - returns true if empty |
Double Ended Queues(deque): Allows for insertions and removals from front and back
- By adding reference to previous node - removals occur in O(1) ATD: StacksDef: | Operations allowed at only one end of the list (top) | | | LIFO: (Last in first out) | Operations: | push() - inserts element at top | | | pop() - removes object at top | | | top() - returns top element without removing it | | | size() - returns number of elements | | | isEmpty() - returns True if empty | Applications | page visited history in web browser | | | JVM - keeps track of chain of active elements (allows for rec) | Performance: | space used: O(n) | | | operations: O(1) | Limitations | max size must be defined prior | | | pushing to a full stack causes implementation specific error |
| | BinarySearchBinarySearch(A[0..N-1], value) {
low = 0
high = N - 1
while (low <= high) {
mid = (low + high) / 2
if (A[mid] > value)
high = mid - 1
else if (A[mid] < value)
low = mid + 1
else
return mid
}
return not_found // value would be inserted at index "low"
} |
Invariants:
value > A[i] for all i < low value < A[i] for all i > high
Worst case performance: O(log n)
Best case performance: O(1)
Average case performance: O(log n) BinarySearch (Recursive)int bsearch(int[] A, int i, int j, int x) {
if (i<j) {
int e = [(i+j)/2];
if (A[e] > x) {
return bsearch(A,i,e-1);
} else if (A[e] < x) {
return bsearch(A,e+1,j);
} else {
return e;
}
} else { return -1; }
} |
Time Complexity: log(base2)(n) Insertion Sort (Iterative)For i ← 1 to length(A) - 1
j ← i
while j > 0 and A[j-1] > A[j]
swap A[j] and A[j-1]
j ← j - 1
end while
end for |
| | Merge-then-sortAlgorithm ListIntersection (A,m, B,n)
Input: Same as before
Output: Same as before
inter ← 0
Array C[m+n];
for i ← 0 to m-1 do C[i] ← A[i];
for i ← 0 to n-1 do C[ i+m ] ← B[i];
C ← sort( C, m+n );
ptr ← 0
while ( ptr < m+n-1 ) do {
if ( C[ptr] = C[ptr+1] ) then {
inter ← inter+1
ptr ← ptr+2
}
else ptr ← ptr+1
}
return inter |
Time Complexity: (m+n) * (log(m+n)) + m + n -1 MergeSort (Recursive)MergeSort (A, p, r) // sort A[p..r] by divide & conquer
if p < r
then q ← ⎣(p+r)/2⎦
MergeSort (A, p, q)
MergeSort (A, q+1, r)
Merge (A, p, q, r) |
Time Complexity: 2T(n/2) + k • n + c’ Primitive Operationsassignment | calling method | returning from method | arithmetic | comparisons of primitive types | conditional | indexing into array | following object reference |
Assume each primitive operation holds the same value = 1 primitive operation |
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