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1995 ACM Scholastic Programming Contest Finals

sponsored by Microsoft ®

Problem C
Variable Radix Huffman Encoding

Huffman encoding is a method of developing an optimal encoding of the symbols in a source alphabet using symbols from a target alphabet when the frequencies of each of the symbols in the source alphabet are known. Optimal means the average length of an encoded message will be minimized. In this problem you are to determine an encoding of the first N uppercase letters (the source alphabet, S1 through SN, with frequencies f1 through fN) into the first R decimal digits (the target alphabet, T1 through TR).

Consider determining the encoding when R=2. Encoding proceeds in several passes. In each pass the two source symbols with the lowest frequencies, say S1 and S,2,, are grouped to form a new "combination letter" whose frequency is the sum of f1 and f2. If there is a tie for the lowest or second lowest frequency, the letter occurring earlier in the alphabet is selected. After some number of passes only two letters remain to be combined. The letters combined in each pass are assigned one of the symbols from the target alphabet. The letter with the lower frequency is assigned the code 0, and the other letter is assigned the code 1. (If each letter in a combined group has the same frequency, then 0 is assigned to the one earliest in the alphabet. For the purpose of comparisons, the value of a "combination letter" is the value of the earliest letter in the combination.) The final code sequence for a source symbol is formed by concatenating the target alphabet symbols assigned as each combination letter using the source symbol is formed. The target symbols are concatenated in the reverse order that they are assigned so that the first symbol in the final code sequence is the last target symbol assigned to a combination letter. The two illustrations below demonstrate the process for R=2.

      Symbol  Frequency                  |  Symbol  Frequency
        A         5                      |  A         7
        B         7                      |  B         7
        C         8                      |  C         7
        D        15                      |  D         7
Pass 1: A and B grouped	       	       	 | Pass 1: A and B grouped
Pass 2: {A,B} and C grouped		 | Pass 2: C and D grouped
Pass 3: {A,B,C} and D grouped		 | Pass 3: {A,B} and {C,D} grouped
Resulting codes: A=110, B=111, C=10, D=0 | Resulting codes: A=00, B=01, C=10, D=11
Avg. length = (3*5+3*7+2*8+1*15)/35=1.91 | Avg. length = (2*7+2*7+2*7+2*7)/28=2.00
When R is larger than 2, R symbols are grouped in each pass. Since each pass effectively replaces R letters or combination letters by 1 combination letter, and the last pass must combine R letters or combination letters, the source alphabet must contain k*(R-1)+R letters, for some integer k. Since N may not be this large, an appropriate number of fictitious letters with zero frequencies must be included. These fictitious letters are not to be included in the output. In making comparisons, the fictitious letters are later than any of the letters in the alphabet.

Now the basic process of determining the Huffman encoding is the same as for the R=2 case. In each pass, the R letters with the lowest frequencies are grouped, forming a new combination letter with a frequency equal to the sum of the letters included in the group. The letters that were grouped are assigned the target alphabet symbols 0 through R-1. 0 is assigned to the letter in the combination with the lowest frequency, 1 to the next lowest frequency, and so forth. If several of the letters in the group have the same frequency, the one earliest in the alphabet is assigned the smaller target symbol, and so forth. The illustration below demonstrates the process for R=3.

      Symbol   Frequency
        A	   5
        B          7
        C          8
        D         15
Pass 1: ? (fictitious symbol), A and B are grouped
Pass 2: {?,A,B}, C and D are grouped
Resulting codes: A=11, B=12, C=0, D=2
Avg. length = (2*5+2*7+1*8+1*15)/35=1.34


The input will contain one or more data sets, one per line. Each data set consists of an integer value for R (between 2 and 10), an integer value for N (between 2 and 26), and the integer frequencies f1 through fN, each of which is between 1 and 999. The end of data for the entire input is the number 0 for R; it is not considered to be a separate data set.


For each data set, display its number (numbering is sequential starting with 1) and the average target symbol length (rounded to two decimal places) on one line. Then display the N letters of the source alphabet and the corresponding Huffman codes, one letter and code per line. The examples below illustrate the required output format.

Sample Input

2 5 5 10 20 25 40
2 5 4 2 2 1 1
3 7 20 5 8 5 12 6 9
4 6 10 23 18 25 9 12

Output for the Sample Input

Set 1; average length 2.10
    A: 1100
    B: 1101
    C: 111
    D: 10
    E: 0

Set 2; average length 2.20
    A: 11
    B: 00
    C: 01
    D: 100
    E: 101

Set 3; average length 1.69
    A: 1
    B: 00
    C: 20
    D: 01
    E: 22
    F: 02
    G: 21

Set 4; average length 1.32
    A: 32
    B: 1
    C: 0
    D: 2
    E: 31
    F: 33

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Last updated September 20, 1999