Searching in dara structure and algorithm

1. Searchinginanarray
CS10003PROGRAMMINGANDDATASTRUCTURES
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2. Searching
Checkif agiven element(calledkey) occursinthe array.
• Example:arrayofstudentrecords; rollno canbethe key.
Twomethodstobediscussed:
a) If thearray elementsareunsorted.
• Linear search
b) If thearray elementsaresorted.
• Binarysearch
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3. BasicConceptofLinearSearch
Basicidea
• Startatthebeginningofthe array.
• Inspectelementsonebyonetoseeif it matchesthekey.
• If amatchis found,returnthearray indexwherethematchwasfound.
• If nomatchis found,aspecialvalue is returned (like –1).
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4. LinearSearch(contd.)
Function linear_search returns the array indexwhereamatchis found.
It returns–1if there is nomatch.
int linear_search (int a[], int size, int key)
{
int pos = 0;
while ((pos < size) && (key != a[pos]))
pos++;
if (pos < size)
return pos;
return -1;
/* Return the position of match */
/* No match found */
}
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5. TimeComplexityofLinearSearch
Ameasureofhowmanybasicoperations analgorithm needstoperform beforeterminating.
Exampleofbasicoperation:match/ comparetwoelements.
• If therearenelementsinthearray:
• Bestcase:
matchfoundinfirst element(1 searchoperation)
• Worstcase:
nomatchfound,ormatchfoundinthe lastelement(nsearchoperations)
• Average case: (n+1)/ 2 searchoperations
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6. Binary Search
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7. BasicConcept
Binarysearchis applicableif the array is sorted.
BasicIdea
• Lookfor the targetinthe middle.
• If youdon’tfindthe key, you canignore half ofthearray, andrepeatthe process withthe
other half.
Ineverystep,wereduce,byafactor oftwo,thenumberofelementstosearchfrom.
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8. TheBasicStrategy
Whatdowewant?
• Plantofind thearray indexbetweenvalues larger andsmaller thankey:
• Lookatthe elementatindex[(L+R)/2].
• Discardonehalf ofthesearchwindowdependingonthe outcomeoftest.
0
<= key > key
x:
L
• Situationwhilesearching:
• Initially, the searchwindowis theentire array, thatis, LandRareinitialized tothe
indicesofthe first andthelastelements.
n-1
R
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9. InitializationandReturnValue
int bin_search (int x[], int size, int key)
{
int L, R, mid;
L = 0; R = size - 1;
while (L != R)
{
mid = (L + R) / 2;
If key
locatio
If not
appears inx[0…size–1],return its
n,possuchthatx[pos]==key.
found,return –1
}
if (key <= x[mid])
else L = mid + 1;
}
if (key == x[L])
return L;
else
return -1;
R = mid;
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10. BinarySearchExamples
bin_search (x, 9, 3); binsearch(x, 9, 2);
-17 -5 3 6 12 21 45 63 50
Trace
0 1 2 3 4 5 6 7 8
Sortedarray
L R M key <= x[M]?
0 8 4 3 <= 12? [True]
0 4 2 3 <= 3? [True]
0 2 1 3 <= -5? [False]
2 2
key == x[L]?
[Loop terminates]
[True]
Wecanmodify thealgorithm bycheckingequalitywithx[mid].
L R M key <= x[M]?
0 8 4 2 <= 12? [True]
0 4 2 2 <= 3? [True]
0 2 1 2 <= -5? [False]
2 2
key == x[L]?
[Loop terminates]
[False]
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11. AnotherVersionofIterative BinarySearch
int bin_search_1 (int x[], int size, int key)
{
int L, R, mid;
L = 0; R = size-1;
while (L <= R)
{
mid = (L + R) / 2;
if (key == x[mid]) return mid;
if (key < x[mid]) R = mid - 1;
else L = mid + 1;
}
return -1;
}
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12. UnsortedvsSortedArraySearch:Where’sthedifference?
Supposethatthearray xhas1000elements.
Linear search
If keyis amemberofx, it wouldrequireabout500comparisons ontheaverage.
Binarysearch
• After 1stcompare,left with500elements.
• After 2ndcompare,left with250elements.
• After atmost10steps,you aredone.
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13. TimeComplexity
If there arenelementsinthearray.
• Numberofiterations required:
log2n
Forn=64(say).
• Initially, list size=64.
• After firstcompare,list size=32.
• After secondcompare,list size=16.
• After thirdcompare,list size=8.
• …
• After sixthcompare,list size=1.
log264 =6
log21024 =10
2k=n,wherekisthe
numberofsteps.
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14. Recursive VersionofBinarySearch
Thealgorithmforbinarysearchdirectlyleadstoarecursiveformulation.
• Thealgorithmis calledrecursively byadjustingtheleft orrightpointers,asapplicable.
• Thebaseconditionis:theelement is found,ortheleft andrightpointerscross.
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15. int binarySearch (int x[], int L, int R, int key)
{
// If the element is present at the middle
int mid;
if (L <= R) {
mid = (L + R) / 2;
if (key == x[mid])
return mid;
if (key < x[mid]) // Look into the left subarray
else
return binarySearch (x, L, mid-1, key);
// Look into the right subarray
return binarySearch (x, mid+1, R, key);
}
// Element is not present in array
return -1;
}
int result = binarySearch (arr, 0, n-1, key);
if (result == -1)
printf ("Key is not present in arrayn");
else
printf("Key is present at index %dn", result);
Returns locationofkeyingivenarray
arr[L …R]if present, otherwise–1
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