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# Questions tagged [summation]

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Section A.1 of the Mathematical Appendix of the CLRS, the third edition, page 1146, contains the following formula stating linearity property of summation applied to $\Theta$ notation: $$\sum_{k=1}^{... 4 votes 1 answer 124 views ### Maximum sum of values in a square grid (one in each row/ column) this is my first post here so bare with me :). What i'm looking for is an algorithm that can find the maximum sum of values in a square grid under the restriction, that you can only pick 1 value from ... 2 votes 1 answer 18 views ### sum of Boolean characters larger degree I was curious if someone knew the answer/reference for the following. So it is well-known that if S\in \{0,1\}^n, then$$ \frac{1}{2^n}\sum_{x\in \{0,1\}^n} (-1)^{\langle S, x\rangle}=1 $$if and ... 1 vote 1 answer 48 views ### Is there a formula for this summation \sum\limits_{k=i}^\mathbb{N}{i}? Is there a formulae for this summation? I wonder this could be not a constant or to say that i * i? \sum\limits_{k=i}^\mathbb{N}{i} 0 votes 1 answer 32 views ### Factorial Formulae proof (from Algorithm Design Manual) I'm going through Algorithm Design Manual and it didn't take long before I hit a proof I don't understand. Can anyone point me in the right direction? From the book: Problem: Prove that \sum_{i=1}^n ... 0 votes 1 answer 18 views ### Efficiently Taking the Sums of all Pairs of Elements from a Set I have a set of numbers and I want to find the sums of all possible pairs of elements. With the set \{1, 2, 3\} for example, I would want \{1+2, 1+3, 2+3\} as my answer. I could do that in n^2 ... 0 votes 1 answer 93 views ### Understanding a summation notation. Sum(j=2 to n) j - 1 I have been reading analysis of insertion sort in the "Introduction to algorithms" and faced a problem with understanding a specific summation notation when the worst case occurs. I know how ... 0 votes 1 answer 54 views ### Asymptotic notation for summations I am struggling to understand why this property of asymptotic notation is true 2 votes 2 answers 109 views ### Closed-form expression for the sums of the rows of the Trinomial triangle There is a question in the second chapter of the book that I'm unable to solve, and unfortunately algorist.com does not provide a rigorous enough solution, or maybe I can't quite understand it. Here ... 0 votes 2 answers 76 views ### Summation and big theta Can anyone suggest a starter book or website where I can find information and examples of putting functions under summation signs into Big Theta. basic Arihmetic , geometric, Quadratic are fine , but ... 4 votes 2 answers 6k views ### Sum of series (n/1) + (n/2) + (n/3) + (n/4) +…….+ (n/n) I wonder if there is a formula to calculate the sum of n/1 + n/2 + n/3 + n/4 + ... + 1. (Integer division) The number n can be as large as 10^12, so a formula or a ... 0 votes 2 answers 275 views ### Estimate the running time of a while-for-while loop i, sq ← 1, 1 while sq < n for j ← 1 to sq k ← 1 while k ≤ j k ← 2 ∗ k i ← i + 1 sq ← i ∗ i I have expressed the running ... 2 votes 0 answers 84 views ### How to parallelize a summation efficiently Say I have an array a[1..n], and I want to output an array s[1..n] with s[i] = a+...+a[i]. What is the best (or at least standard) way to do so in parallel? The way I can think of doing it, given m ... 0 votes 0 answers 61 views ### Reconstructing an Array via Time-Intensive Subset Queries I am trying to design an algorithm for a problem, and the following is an auxiliary problem for which a good solution would imply a faster algorithm for the original problem. I am given access to an ... 0 votes 3 answers 144 views ### Find the smallest group of numbers with sum bigger then X Given a list of numbers S where 0 < s_i < 100, find the minimum sum group of numbers with a sum bigger than X. Each number can be used multiple times. Ex: for S = [3,4.1], X = 10 the ... -1 votes 2 answers 79 views ### What would be the complexity for this relation? The function: for (int i=1; i<n²; i++) for (int j=i; j<n; j++) print(j) Putting it into a relation, I got: C1 + \sum_{i=1}^{n^2}(C2+... 2 votes 1 answer 789 views ### How to solve recurrence T(n) = 5T(\frac{n}{2}) + n^2\lg^2 n I have tried solve the recurrence T(n) = 5T(\frac{n}{2}) + n^2\lg^2 n using substitution. Apparently, it is exact for some n and the order of the general solution can be found from this exact ... 0 votes 1 answer 71 views ### Calculate the number of iterations in unusual nested loop I am trying to calculate the number of iterations of a sequence of nested loops of the form: \begin{equation} N = \sum_{j=0}^{j_T} \sum_{k=0}^{j} \sum_{l=0}^{k} \sum_{n=n_0}^{n_T} 1 \end{equation} ... 1 vote 2 answers 553 views ### Summation of asymptotic notation How can we solve summation of asymptotic notations like given below:$$ \sum_{k=1}^{n-1} O(n). $$4 votes 2 answers 253 views ### Why is \sum_{i=1}^n O(i) not the same as O(1)+O(2)+\dots+O(n)? The well-known textbook Introduction to Algorithms ("CLRS", 3rd edition, chapter 3.1) claims the following:$$ \sum_{i=1}^n O(i)  is not the same as (I'm not using DNE because the book explicitly ... 