Transform and Conquer - Instance Simplification

Transform and Conquer - Instance Simplification

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原理(Principles)
Try to make the problem easier through some pre-processing, typically sorting. We can pre-sort input to speed up.

Uniqueness Checking - Brute Force

function UniquenessChecking(A[0..n-1])
    for i ⟵ 0 to n-2 do
        for j ⟵ i+1 to n-1 do
            if A[i] = A[j] then
                return false
    return true

Time complexity: Θ(n^2).

Uniqueness Checking - Presorting

function UniquenessChecking(A[0..n-1])
    Sort(A[0..n-1])
    for i ⟵ 0 to n-2 do
        if A[i]=A[i+1] then
            return false
    return true

Time Complexity: O(nlogn). (If we use some algorithms based on key comparison which belong to O(nlogn)).

Mode
A mode is a list or array element which occurs most frequently in the list or array. For example,
42, 78, 13, 13, 57, 42, 57, 78, 13, 98, 42, 33
The element 13 and 43 are modes.

Computing Mode - Brute Force

function mode(A[0..n-1])
    maxfreq ⟵ 0
    count ⟵ 0
    for i ⟵ 0 to n-1 do
        for j ⟵ 0 to n-1 do
            if A[i] = A[j] then
                count++
        if count > maxfreq then
            maxfreq ⟵ count
            mode ⟵ A[i]
        count ⟵ 0
    return mode

Time Complexity: Θ(n^2).

Computing Mode - Presorting

function Mode(A[0..n-1])
    Sort(A[0..n-1])
    i ⟵ 0
    maxfreq ⟵ 0
    while i < n do
        runlength ⟵ 1
        while i + runlength < n and A[i+runlength] = A[i] do
            runlength ⟵ run length + 1
        if runlength > maxfreq then
            maxfreq ⟵ runlength
            mode ⟵ A[i]
        i ⟵ i + runlength 
    return mode

Time Complexity: O(nlogn).

Sequential Search

function SequentialSearch(A[0..n-1], k)
    i ⟵ 0
    while A[i] ≠ k and i < n do
        i ⟵ i + 1
    if i=n then
        return -1
    else
        return i

Time Complexity: Θ(n).

Binary Search

function BinarySearch(A[0..n-1],K)
    lo ⟵ 0
    hi ⟵ n-1
    while lo<=hi do
        m ⟵ ⎣(hi+lo)/2⎦
        if A[m]=K 
            return m
        else if A[m]>K then
            hi ⟵ m-1
        else A[m]<k then 
            lo ⟵ m+1
    return  -1

Time Complexity: Θ(logn).

寫在最後的話(PS)
Transform and conquer is one of algorithmic design techniques. In some case we can use this technique to simplify problems.
Welcome questions always and forever.