The primary topics in this part of the specialization are: greedy algorithms (scheduling, minimum spanning trees, clustering, Huffman codes) and dynamic programming (knapsack, sequence alignment, optimal search trees).

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From the course by Stanford University

Greedy Algorithms, Minimum Spanning Trees, and Dynamic Programming

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The primary topics in this part of the specialization are: greedy algorithms (scheduling, minimum spanning trees, clustering, Huffman codes) and dynamic programming (knapsack, sequence alignment, optimal search trees).

From the lesson

Week 3

Huffman codes; introduction to dynamic programming.

- Tim RoughgardenProfessor

Computer Science

To make sure that Huffman's greedy algorithm is clear,

Â let's go through a slightly larger, more complicated example.

Â So let's work with a six character alphabet.

Â Let's call the letters A, B, C, D, E, F, and let's assume

Â that we're given the weights 3, 2, 6 and 8, 2, 6 for these six characters.

Â Remember, this problem is well-defined even if the weights don't add up to 1.

Â If you prefer working with actual probabilities,

Â feel free to divide these six numbers by 27.

Â In the first step of Huffman's greedy algorithm,

Â we find the letters that have the smallest weights, the smallest frequencies.

Â So in this example, that would be the letters B and E,

Â both of them have weight 2.

Â Now what we do is we merge these two letters into a single meta letter,

Â in effect committing right now to having B and E be siblings in the final tree.

Â After this merger, our alphabet is down to five symbols, the symbols B and

Â E being replaced by the merged symbol BE.

Â And the weight of BE is the sum of the weights of B and E, namely, 4.

Â We can imagine our tree slowly taking shape through these iterations.

Â So after step 1, we know that B and E are going to be siblings and we know that just

Â A, C, D, and F are going to be leaves, that's all we know thus far.

Â In the next iteration, we again look for

Â the two symbols that have the smallest weight.

Â And here the smallest weight symbol is A.

Â It has weight 3.

Â And the runner-up is the merged symbol B sub E.

Â Its combined weight is 4, and that's second overall for these five symbols.

Â So in this step, we merge A with B E.

Â Now our alphabet is down to four symbols, the merged symbol, A, B, E,

Â which has cumulative weight 7, and then the original symbols, C, D, and F,

Â which have their original weights, 6, 8, and 6.

Â As far as our tree, we've now committed to the symbol A appearing as an uncle

Â of the siblings B and E.

Â And again, C, D,

Â and F, we just know they're leaves somewhere in the final tree.

Â In step three,

Â we're going to again pick the two symbols that have the smallest weights.

Â In this case, the two symbols with the smallest weight are C and

Â F, each of weight 6.

Â In our new alphabet, we still have our symbol ABE, it still has weight 7.

Â We still have the symbol D, it still has weight 8.

Â But now we have a new merged symbol CF, and its new weight is 12.

Â As far as our tree, in addition to the information we already knew,

Â we're now committing to having C and F be siblings in the final tree.

Â In step four, we merge the two symbols with the smallest weight.

Â So that would be ABE with its weight of seven and D with its weight of eight.

Â So this leaves us with only two symbols, ABDE and CF.

Â And now we know what both of these subtrees of the root of the final tree

Â are going to look like.

Â Now that we're down to two symbols,

Â the only thing we can do is fuse these two symbols into one,

Â fuse these two subtrees into a single one by uniting them under a common root.

Â That gives us the following final output of Huffman's algorithm.

Â What prefix-free code does this tree correspond to?

Â Well, as usual, let's label all of the left branches with zero and

Â all of the right branches with ones.

Â And now, as usual, the encoding of a character is just the symbols of zeros and

Â ones that you see when you traverse a path from the root to that leaf.

Â So for example, A will be encoded with 000,

Â B with 0010, C with 10, D with 01,

Â E with 0011, and F with 11.

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