Uddiṣṭa: From a Pattern to Its Row Number

Worked by hand: G G G L G L

Row forty-one, which is the row [Binary Number Systems, and Exactly Where Piṅgala's Order Agrees] reached by arithmetic. The two methods agree.

Why the rule is right

Write the row number, counted from one, as one plus the pattern's binary value with laghu as 1 and the leftmost syllable in the units place. Then the rule is Horner's method for evaluating that expression from the high-order end, which for this direction of writing means from the right.

Let y stand for the running number after some syllables have been taken. The step "double, and subtract one at a guru" is the same as "double, and add one at a laghu, then subtract one". Which is to say: it evaluates twice the previous value, plus the current bit, minus one. Unrolled from a start of one, that is exactly one plus the binary value.

The short version, which is the one to write in an examination: the rule is Horner's evaluation of the pattern as a binary number, shifted by one because rows are counted from one and numbers from zero.

The rule, run

def uddista(pattern):
    """8.26 pratilomagunam dvirladyam, with 8.27 tato'pyekam jahyat.

    Take the syllables from the LAST to the first. Start at one. At each
    syllable double; and if that syllable is a guru, subtract one.
    """
    y = 1
    for ch in reversed(pattern):
        y *= 2
        if ch == "G":
            y -= 1
    return y

for p in ("GGGGGG", "LGLGGG", "GGGLGL", "LLLLLL", "GGG", "LLL"):
    print("%-8s row %d" % (p, uddista(p)))

GGGGGG   row 1
LGLGGG   row 6
GGGLGL   row 41
LLLLLL   row 64
GGG      row 1
LLL      row 8

Four lines of code for a rule stated in two sūtras. The whole content is y = 2*y with a conditional y -= 1.

The cost

One pass over the pattern, one doubling and at most one subtraction per syllable. So the work is proportional to n, the number of syllables, and does not depend on how large the row number is.

Compare the alternative: generating the table and searching it costs n times 2 to the power n. For the jagatī that is 49,152 symbol operations against 12. This is the same improvement in kind that an index gives over a scan, and it is worth saying in exactly those words, because that is what MU's phrase "index retrieval mechanisms" is pointing at.

What it does NOT mean

It does not mean the pattern is validated. Give the rule a pattern of the wrong length and it will cheerfully return a number. Nothing in the text checks the input, and a program should.

It does not mean rows are counted from zero. They are counted from one throughout the tradition, which is why the rule starts at one rather than at zero. Mixing the conventions is the commonest arithmetic mistake in this topic.

It does not work for another order. Later prosodists use the reverse order, and uddiṣṭa for that order is a different rule. Applying this one to a table built the other way gives a wrong answer silently.

Quick revision

• Uddiṣṭa: from a pattern to its row number.
• 8.26 pratilomaguṇaṃ dvirlādyam, reverse order, doubling; 8.27 tato'pyekaṃ jahyāt, and subtract one, which Halāyudha applies at a guru.
• The rule: start at one, take syllables from last to first, double each time, subtract one at a heavy syllable.
• Worked: L G L G G G is row 6; G G G L G L is row 41.
• It is Horner's evaluation of the pattern as a binary number, offset by one because rows count from one.
• Cost proportional to n, against n times 2 to the power n for generate and search.

Test yourself

1. Find the row number of G L L G by the rule, showing the running value.

Syllables from the last: G, L, L, G. Start 1. G: double to 2, subtract one, 1. L: double to 2. L: double to 4. G: double to 8, subtract one, 7. Row seven.

2. Why do trailing heavy syllables leave the running number unchanged?

Because doubling one and subtracting one gives one again. In the binary reading they are leading zeros, which contribute nothing to a number's value.

3. State the modern description of the rule in one sentence.

It is Horner's method for evaluating the pattern as a binary number with laghu as one and the leftmost syllable in the units place, offset by one because rows are counted from one.

4. Give the cost of uddiṣṭa and of the obvious alternative, for a twelve-syllable metre.

Uddiṣṭa is twelve doublings, one per syllable. Generating the table and searching it is 4096 rows of twelve symbols, about 49,000 symbol operations.