3/06: 1. introduction to the maximum subsequence sum problem
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1.0 problem statement -- english specification: 

    find the maximum subsequence sum of a 0-terminated input sequence
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1.1 example:

for the input sequence:

   +---+----+----+----+---+
   | 6 | -2 | -1 | -2 | 7 | 0
   +---+----+----+----+---+

the subsequence sums are:

 0 []                        empty subsequence
                             
   [ 6 ]                     subsequences ending on 6
                             
   [ 6 + -2 ]                subsequences ending on (first) -2
       [ -2 ]                
                             
   [ 6 + -2 + -1 ]           subsequences ending on -1
       [ -2 + -1 ]           
            [ -1 ]           
                             
   [ 6 + -2 + -1 + -2 ]      subsequences ending on (second) -2
       [ -2 + -1 + -2 ]      
            [ -1 + -2 ]      
                 [ -2 ]      
                             
   [ 6 + -2 + -1 + -2 + 7 ]  subsequences ending on 7
       [ -2 + -1 + -2 + 7 ]  
            [ -1 + -2 + 7 ]  
                 [ -2 + 7 ]  
                      [ 7 ]  
                             
                             stopping value 0 is *not* part of the sequence

we want the maximum of the sums above
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1.2 elaboration of english specification (terminology review & introduction):


0-terminated input sequence:

+ an *input sequence* is a sequence of numbers typed in the user 
  (usually in response to input prompts)

+ a *0-terminated* input sequence is an input sequence containing no 0s,
  where the user indicates the sequence is over by entering in the
  *stopping value* 0, which is thus *not* considered to be part of the
  input sequence


subsequence:

+ a *subsequence* ("subsequence" = "subset" + "sequence") of a vector $s$ 
  is a sequence $s(i:j)$ of consecutive elements of $s$ in the same order.

+ above, we did not put any conditions on $i$ and $j$:
  when conditions (e.g. bounds) on positions are (partly) omitted,
  the omission is shorthand for "can be any ``legal'' value",
  i.e. "allowed to vary over the largest ``reasonable'' set of values".

  note: if you think we accidentally omitted conditions, then ask!

        it is possible we intentionally omitted them,
        but it is also possible we accidentally omitted them.

+ in this case, the largest set of legal values that would not produce an
  error is:

    $i>j$ or $1<=i,j<=length(s)$

  where $1<=i,j<=length(s)$ is shorthand for
 
    $1<=i$ and $1<=j$ and $i<=length(s)$ and $j<=length(s)$

+ when $i>j$, $i:j = []$, so $s(i:j) = s([]) = []$ is the empty sequence.

+ thus, the empty sequence $[]$ is a subsequence of every sequence,
  including itself, just as the empty set {} is a subset of every
  set, including itself.


subsequence sum

+ by itself, the phrase "subsequence sum" means 
  "the sum $sum(s(i:j))$ of a subsequence $s(i:j)$ of some sequence $s$"

+ since $s,i,j$ are omitted, again this is shorthand for 
  "allowed to vary as widely as possible (without error)"

+ "subsequence sum of a( 0-terminated) input sequence"
  does specify the sequence $s$, but since $i,j$ are omitted,
  that means $i,j$ are allowed to vary as much as possible


"find the maximum <blah>"

+ this means "find the maximum value that $blah$ can attain for all legal
  values of the associated variables (explicit and implicit)".
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1.3 problem statement -- technical specification: 

  compute $max sum(s(i:j))$ for a given sequence $s$ of input values
           i,j            

+ the notation

      max f(i,j) = "max of $f(i,j)$ over all legal values of $i$ and $j$"
      i,j
  
  is just like

      ----
      \
       )    f(i,j) = "sum of $f(i,j)$ over all legal values of $i$ and $j$"
      /
      ----
      i,j                            __
                                    \
  i.e. an operation (e.g. $max$ or $ )  $) applied to values $f(i,j)$
                                    /__

       as $i,j$ range over all allowed values.

+ again, the lack of conditions on $i$ and $j$ means they are allowed
  to vary as much as possible (so long as there is no error)

+ since the empty sequence $[]$ is always a subsequence and 
  has sum $sum([]) = 0$, this tells us the maximum subsequence sum:
  + is 0 for the empty sequence $s = []$
  + is greater than or equal to zero (i.e. is always non-negative)
    (since the empty sequence is a subsequence, the maximum subsequence sum
    is greater than or equal to the subsequence sum for the empty subsequence)
  + is 0 if every element of $s$ is negative
    + note: $s = []$ is a special case of "every element of $s$ is negative"

note: 

+ mathematically, $max([]) = -inf$
+ but matlab returns $[]$ for $max([])$ -- grr!
+ in lecture, someone asked "how can $sum(T)$ be bigger than $max(T)$?"

     ("how can the sum of a sequence or set be bigger than 
     the max of the same sequence or set?"),

  this is actually not so surprising:
  
  + consider $T = [1 1 1 1]$: then $sum(T) = 4 > 1 = max(T)$.
  
  + consider $T = []$: then $sum(T) = 0 > -inf = max(T)$.
  
  + of course, it is not true that $sum(T)$ is always bigger than $max(T)$ --
    consider $T = [-1 -1]$: then $sum(T) = -2 < -1 = max(T)$.

+ also, observe that the sums and max are being computed over different sets:

  + the sums are of subsequences of the input sequence
  + conceptually, the $max$ operation in the definition:
    + IS the max of a set of the *sums* (of subsequences)
    + IS NOT the max of a subsequence or set of subsequences
  + e.g. if $s = [3 7]$
    + conceptually, the max is computed on the set
      { sum(s([]), sum(s(1:1)), sum(s(2:2)), sum(s(1:2)) }
      = { sum([]), sum([3]), sum([7]), sum([3 7]) }
      = { 0, 3, 7, 10 }
    + conceptually, the max is not computed on any of these sequences:
      + the sequence s([]) = []
      + the sequence s(1:1) = [3]
      + the sequence s(2:2) = [7]
      + the sequence s(1:2) = [3 7]
  + e.g. if $s = []$
    + conceptually, the max is computed on the set
      { sum(s([]) } = { sum([]) } = { 0 }, yielding the answer 0
    + conceptually, the max is not computed on the sequence $s = s([]) = []$,
      which would yield $-inf$
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1.4. goal: a short and efficient solution to max subsequence sum

+ i.e. code that is *concise, *small*, and *fast*

+ *concise* means the code we write/type is small

+ *small* means the amount of computer memory required 
  --both to store the program and to store data--
  is low, e.g. 3 numbers versus $n$ numbers.

+ *fast* means the amount time to run the program is short,
  e.g. $2n$ steps versus $n^3/6$ or $n^2$ steps.