# Editing Prefix sum array and difference array

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To solve this, we will first transform array <math>S</math> into its prefix sum array <math>P(0,S)</math>. Notice that the sum of each contiguous subsequence <math>S_i + S_{i+1} + S_{i+2} + ... + S_{j-1}</math> corresponds to the difference of two elements of <math>P</math>, that is, <math>P_j - P_i</math>. So what we want to find is the number of pairs <math>(i,j)</math> with <math>P_j - P_i = 47</math> and <math>i < j</math>. (Note that if <math>i > j</math>, we will instead get a subsequence with sum -47.) | To solve this, we will first transform array <math>S</math> into its prefix sum array <math>P(0,S)</math>. Notice that the sum of each contiguous subsequence <math>S_i + S_{i+1} + S_{i+2} + ... + S_{j-1}</math> corresponds to the difference of two elements of <math>P</math>, that is, <math>P_j - P_i</math>. So what we want to find is the number of pairs <math>(i,j)</math> with <math>P_j - P_i = 47</math> and <math>i < j</math>. (Note that if <math>i > j</math>, we will instead get a subsequence with sum -47.) | ||

− | However, this is quite easy to do. We sweep through <math>P</math> from left to right, keeping a [[map]] of all elements of <math>P</math> we've seen so far, along with their frequencies; and for each element <math>P_j</math> we count the number of times <math>P_j - | + | However, this is quite easy to do. We sweep through <math>P</math> from left to right, keeping a [[map]] of all elements of <math>P</math> we've seen so far, along with their frequencies; and for each element <math>P_j</math> we count the number of times <math>P_j - 48</math> has appeared so far, by looking up that value in our map; this tells us how many contiguous subsequences ending at <math>S_{j-1}</math> have sum 47. And finally, adding the number of contiguous subsequences with sum 47 ending at each entry of <math>S</math> gives the total number of such subsequences in the array. Total time taken is <math>O(N)</math>, if we use a [[hash table]] implementation of the map. |

==Use of difference array== | ==Use of difference array== | ||

− | The difference array is used to keep track of an array when ranges of said array can be updated all at once. If we have array <math>A</math> and add an increment <math>k</math> to elements <math>A_i, A_{i+1}, ..., A_{ | + | The difference array is used to keep track of an array when ranges of said array can be updated all at once. If we have array <math>A</math> and add an increment <math>k</math> to elements <math>A_i, A_{i+1}, ..., A_{j-1}</math>, then notice that <math>D_0, D_1, ..., D_{i-2}</math> are not affected; that <math>D_{i-1} = A_i - A{i-1}</math> is increased by <math>k</math>; that <math>D_i, D_{i+1}, ..., D_{j-2}</math> are not affected; that <math>D_{j-1} = A_j - A_{j-1}</math> is decreased by <math>k</math>; and that <math>D_j, D_{j+1}, ...</math> are unaffected. Thus, if we are required to update many ranges of an array in this manner, we should keep track of <math>D</math> rather than <math>A</math> itself, and then integrate at the end to reconstruct <math>A</math>. |

===Example: Wireless (CCC)=== | ===Example: Wireless (CCC)=== | ||

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After the prefix sum array has been computed, we can use it to add together any rectangle in <math>A</math>, that is, all the elements with their indices in <math>[x_1,x_2), [y_1,y_2)</math>. To do so, we first observe that <math>P_{x_2,y_2} - P_{x_1,y_2}</math> gives the sum of all the elements with indices in the box <math>[x_1, x_2) \times [0, y_2)</math>. Then we similarly observe that <math>P_{x_2,y_1} - P_{x_1,y_1}</math> corresponds to the box <math>[x_1, x_2) \times [0, y_1)</math>. Subtracting gives the box <math>[x_1, x_2) \times [y_1, y_2)</math>. This gives a final formula of <math>P_{x_2,y_2} - P_{x_1,y_2} - P_{x_2,y_1} + P_{x_1,y_1}</math>. | After the prefix sum array has been computed, we can use it to add together any rectangle in <math>A</math>, that is, all the elements with their indices in <math>[x_1,x_2), [y_1,y_2)</math>. To do so, we first observe that <math>P_{x_2,y_2} - P_{x_1,y_2}</math> gives the sum of all the elements with indices in the box <math>[x_1, x_2) \times [0, y_2)</math>. Then we similarly observe that <math>P_{x_2,y_1} - P_{x_1,y_1}</math> corresponds to the box <math>[x_1, x_2) \times [0, y_1)</math>. Subtracting gives the box <math>[x_1, x_2) \times [y_1, y_2)</math>. This gives a final formula of <math>P_{x_2,y_2} - P_{x_1,y_2} - P_{x_2,y_1} + P_{x_1,y_1}</math>. | ||

− | In the <math>n</math>-dimensional case, to sum the elements of <math>A</math> with indices in the box <math>[x_{1,0}, x_{1,1}) \times [x_{2,0}, x_{2,1}) \times ... \times [x_{n,0}, x_{n,1})</math>, we will use the formula <math>\sum_{k_1=0}^1 \sum_{k_2=0}^1 ... \sum_{k_n=0}^1 | + | In the <math>n</math>-dimensional case, to sum the elements of <math>A</math> with indices in the box <math>[x_{1,0}, x_{1,1}) \times [x_{2,0}, x_{2,1}) \times ... \times [x_{n,0}, x_{n,1})</math>, we will use the formula <math>(-1)^n \sum_{k_1=0}^1 \sum_{k_2=0}^1 ... \sum_{k_n=0}^1 P_{x_{1,k_1}, x_{2,k_2}, ..., x_{n,k_n}}</math>. We will not state the proof, instead noting that it is a form of the inclusion–exclusion principle. |

====Example: Diamonds (BOI)==== | ====Example: Diamonds (BOI)==== | ||

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We will simply define the difference array <math>D</math> to be the array whose prefix sum array is <math>A</math>. This means in particular that <math>D_{i,j}</math> is the sum of elements of <math>D</math> with indices in the box <math>[i,i+1) \times [j,j+1)</math> (this box contains only the single element <math>D_{i,j}</math>). Using the prefix sum array <math>A</math>, we obtain <math>D_{i,j} = A_{i+1,j+1} - A_{i,j+1} - A_{i+1,j} + A_{i,j}</math>. In general: | We will simply define the difference array <math>D</math> to be the array whose prefix sum array is <math>A</math>. This means in particular that <math>D_{i,j}</math> is the sum of elements of <math>D</math> with indices in the box <math>[i,i+1) \times [j,j+1)</math> (this box contains only the single element <math>D_{i,j}</math>). Using the prefix sum array <math>A</math>, we obtain <math>D_{i,j} = A_{i+1,j+1} - A_{i,j+1} - A_{i+1,j} + A_{i,j}</math>. In general: | ||

− | :<math>D_{i_1, i_2, ..., i_n} = \sum_{k_1=0}^1 \sum_{k_2=0}^1 ... \sum_{k_n=0}^1 | + | :<math>D_{i_1, i_2, ..., i_n} = (-1)^n \sum_{k_1=0}^1 \sum_{k_2=0}^1 ... \sum_{k_n=0}^1 A_{i_1+k_1,i_2+k_2,...,i_n+k_n}</math>. |

Should we actually need to ''compute'' the difference array, on the other hand, the easiest way to do so is by reversing the computation of the prefix sum array: | Should we actually need to ''compute'' the difference array, on the other hand, the easiest way to do so is by reversing the computation of the prefix sum array: | ||

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====Example: The Cake is a Dessert==== | ====Example: The Cake is a Dessert==== | ||

− | {{Problem|cake|The Cake is a Dessert}} from the | + | {{Problem|cake|The Cake is a Dessert}} from the Woburn–Waterloo 2011 Mock CCC is a straightforward application of the difference array in two dimensions. |

==A word on the dynamic case== | ==A word on the dynamic case== | ||

− | The dynamic version of the problem that the prefix sum array is intended to solve requires us to be able to carry out an intermixed sequence of operations, where each operation either changes an element of <math>A</math> or asks us to determine the sum <math>A_0 + A_1 + ... + A_{i-1}</math> for some <math>i</math>. That is, we need to be able to compute entries of the prefix sum array of an array that is changing (dynamic). If we can do this, then we can do range sum queries easily on a dynamic array, simply by taking the difference of prefix sums as in the static case. This is a bit trickier to do, but is most easily and efficiently accomplished using the [[binary indexed tree]] data structure, which | + | The dynamic version of the problem that the prefix sum array is intended to solve requires us to be able to carry out an intermixed sequence of operations, where each operation either changes an element of <math>A</math> or asks us to determine the sum <math>A_0 + A_1 + ... + A_{i-1}</math> for some <math>i</math>. That is, we need to be able to compute entries of the prefix sum array of an array that is changing (dynamic). If we can do this, then we can do range sum queries easily on a dynamic array, simply by taking the difference of prefix sums as in the static case. This is a bit trickier to do, but is most easily and efficiently accomplished using the [[binary indexed tree]] data structure, which allows both queries and updates in <math>O(2^n \log(m_1) \log(m_2) ... \log(m_n))</math> time, where <math>n</math> is the number of dimensions and each <math>m</math> is a dimension. In the most commonly encountered one-dimensional case, this is just logarithmic in the size of the array. |

− | The dynamic version of the problem that the difference array is intended to solve is analogous: we want to be able to increment entire ranges of an array, but at any point we also want to be able to determine any element of the array. Here, we simply use a binary indexed tree for the difference array, and compute prefix sums whenever we want to access an element. | + | The dynamic version of the problem that the difference array is intended to solve is analogous: we want to be able to increment entire ranges of an array, but at any point we also want to be able to determine any element of the array. Here, we simply use a binary indexed tree for the difference array, and compute prefix sums whenever we want to access an element. (Note that this incurs an additional factor of <math>2^n</math> in the running time of the update operation.) |

− | On the other hand, if we combine the two, that is, we wish to be able to increment ranges of some array <math>A</math> as well as query prefix sums of <math>A</math> in an intermixed sequence, matters become more complicated. Notice that in the static case we would just track the difference array of <math>A</math> and then integrate twice at the end. In the dynamic case, however, we need to use a [[segment tree]] with lazy propagation. This will give us <math>O(2^n \log(m_1) \log(m_2) ... \log(m_n))</math> time on both the query and the update, but is somewhat more complex and slower than the binary indexed tree | + | On the other hand, if we combine the two, that is, we wish to be able to increment ranges of some array <math>A</math> as well as query prefix sums of <math>A</math> in an intermixed sequence, matters become more complicated. Notice that in the static case we would just track the difference array of <math>A</math> and then integrate twice at the end. In the dynamic case, however, we need to use a [[segment tree]] with lazy propagation. This will give us <math>O(2^n \log(m_1) \log(m_2) ... \log(m_n))</math> time on both the query and the update, but is somewhat more complex and slower than the binary indexed tree. |

[[Category:Pages needing diagrams]] | [[Category:Pages needing diagrams]] |