Show that for one-dimensional cellular automata, CA-Predecessor is in P. Hint: think about using recursion on a one-bit-shorter prefix of the input configuration. Suppose that the CA follows the rule that the configuration shrinks by 2 cells on each update, since the leftmost cell has no left neighbor and the rightmost cell has no right neighbor. Therefore, on input a configuration of length n, we are asking whether there is a predecessor configuration of length n + 2.   PROVIDE A STEP BY STEP EXPLAINATION SHOWING TIME COMPLEXITY

The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S berepresented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) ofINSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtainedfrom insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n)for i = 1 to n   determine j such that i ∈ ??   if j ≠ m + 1      extracted[j] = i       let L be the smallest value greater than j for which KL exists      KL = KL U Kj, destroying ????return extracted (1) Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where eachnumber stands for its…

The off-line minimum problem maintains a dynamic set T of elements from the domain {1, 2,...,n}under the operations INSERT and EXTRACT-MIN. A sequence S of n INSERT and m EXTRACT-MIN calls are given, where each key in {1, 2,...,n} is inserted exactly once. Let a sequence S berepresented by I1 , E, I2, E, ... , E, Im+1 , where each Ij stands for a subsequence (possibly empty) ofINSERT and each E stands for a single EXTRACT-MIN. Let Kj be the set of keys initially obtainedfrom insertions in Ij. The algorithm to build an array extracted[1..m], where for i = 1, 2, ..., m,extracted[i] is the key returned by the ith EXTRACT-MIN call is given below: Off-Line-Minimum(m, n)for i = 1 to n   determine j such that i ∈ Kj   if j ≠ m + 1      extracted[j] = i       let L be the smallest value greater than j for which KL exists      KL = KL U Kj, destoying Kjreturn extracted   Given the operation sequence 9, 4, E, 6, 2, E, E, 5, 8, E, 1, 7, E, E, 3; where eachnumber stands for its insertion.…

Let x[n] = {1,2,3}. Let y[n] = {1, 2, 3, 4, 5). Circularly convolve x and y where N = 6. Develop the flow-graph of N = 2 x 2 point Decimation-in-time FFT algorithm.