18 1818 18102010-02-08 15:26:14 / rev 470cc001cd97+track whose ‘rings’ lie 1.6 µm apart. Along the track, the pits lie 0.9 µmapart. So, the spacing is between 0.9 and 1.6 µm; if you want just onevalue, let it be the midpoint, 1.3 µm. We made a tasty pudding!Problem 1.3 Robust additionThe text offered addition as an example of intelligent redundancy: We oftenverify an addition by by redoing the sum from bottom to top. Analyze thispractice using simple probability models. Is it indeed an example of intelligentredundancy?Problem 1.4 Intelligent redundancyThink of and describe a few real-life examples of intelligent redundancy.1.3 Theory 2: Tree representationsTasty though the estimation pudding may be, its recipe is long and de-tailed. It is hard to follow – even for its author. Although I wrote theanalysis, I cannot quickly recall all its pieces; rather, I must remind my-self of the pieces by looking over the text. As I do, I am reminded thatsentences, paragraphs, and pages do not compactly represent a divide-and-conquer estimate.Linear, sequential information does not match the estimate’s structure. Itsstructure is hierarchical – with answers constructed from solving smallerproblems, which might be constructed from even solving still smallerproblems – and its most compact representation is as a tree.capacity, areacapacity areaAs an example, let’s construct the tree representing theelaborate divide-and-conquer estimate for a CDROM’s pitspacing (Section 1.1). The tree’s root is ‘capacity, area’, atwo-word tag reminding us of the method underlying theestimate. The estimate dissolves into finding two quanti-ties – the capacity and area – so the tree’s root sprouts two branches.Of the two new leaves, the area is easy to estimate without explicitlysubdividing into smaller problems, so the ‘area’ node remains a leaf. Toestimate the capacity – rather, to estimate the capacity reliably – we usedintelligent redundancy: (1) looking on a CDROM box; and (2) estimatinghow many bits are required to represent the music that fits on an audio18 1818 18102010-02-08 15:26:14 / rev 470cc001cd97+track whose ‘rings’ lie 1.6 µm apart. Along the track, the pits lie 0.9 µmapart. So, the spacing is between 0.9 and 1.6 µm; if you want just onevalue, let it be the midpoint, 1.3 µm. We made a tasty pudding!Problem 1.3 Robust additionThe text offered addition as an example of intelligent redundancy: We oftenverify an addition by by redoing the sum from bottom to top. Analyze thispractice using simple probability models. Is it indeed an example of intelligentredundancy?Problem 1.4 Intelligent redundancyThink of and describe a few real-life examples of intelligent redundancy.1.3 Theory 2: Tree representationsTasty though the estimation pudding may be, its recipe is long and de-tailed. It is hard to follow – even for its author. Although I wrote theanalysis, I cannot quickly recall all its pieces; rather, I must remind my-self of the pieces by looking over the text. As I do, I am reminded thatsentences, paragraphs, and pages do not compactly represent a divide-and-conquer estimate.Linear, sequential information does not match the estimate’s structure. Itsstructure is hierarchical – with answers constructed from solving smallerproblems, which might be constructed from even solving still smallerproblems – and its most compact representation is as a tree.capacity, areacapacity areaAs an example, let’s construct the tree representing theelaborate divide-and-conquer estimate for a CDROM’s pitspacing (Section 1.1). The tree’s root is ‘capacity, area’, atwo-word tag reminding us of the method underlying theestimate. The estimate dissolves into finding two quanti-ties – the capacity and area – so the tree’s root sprouts two branches.Of the two new leaves, the area is easy to estimate without explicitlysubdividing into smaller problems, so the ‘area’ node remains a leaf. Toestimate the capacity – rather, to estimate the capacity reliably – we usedintelligent redundancy: (1) looking on a CDROM box; and (2) estimatinghow many bits are required to represent the music that fits on an audioGlobal comments 1Global commentsAre nodes and leaves the same thing?A Node refers to each individual "quantity" that we are finding. A leaf however is a nodethat does not sprout a new branch. In this tree, ’capacity’ is a node while ’sample rate’is a leaf.lateral movement represents "intelligent" redundancy while longitudinal movement rep-resents divide-and-conquer. I like the trees because they are a more visual way of show-ing/remembering where numbers came from...though I agree that it would be nice if thelongitudinal movements were labeled with equations or something of that sort. I can seethese trees becoming quite complicated for other examples thoughI thought this section was written pretty clearly. The diagrams make it easy to grasp theconcept. Perhaps the wording could be improved. Like someone else said earlier, the first2 paragraphs are awkwardly worded. Other than that, I have no real comments to add! Ihope that’s not a bad thing.I definitely like the visualization of the method. It makes the thought process more clearfor people who like pictures.I don’t think this actually helps in estimation. I feel like the difficulty in a lot of whatwe’re doing is figuring out exactly how we’re going to estimate something, and I feel likethese trees, although new (so it may take some getting used to), maybe just be a wastedway of organizing something that we should be able to handle at any mildly high levelof estimation.I disagree. While many approximations that we’ve done have been simple enough thatthis is not necessary, I can imagine more complex ones for which this would be useful.However, much of its use is in looking back on the work you’ve done when it’s no longerfresh in your mind. You can follow your own logic (or explain it to others) very easilywith a diagram of this type. It also makes errors easier to find and fix.18 1818 18102010-02-08 15:26:14 / rev 470cc001cd97+track whose ‘rings’ lie 1.6 µm apart. Along the track, the pits lie 0.9 µmapart. So, the spacing is between 0.9 and 1.6 µm; if you want just onevalue, let it be the midpoint, 1.3 µm. We made a tasty pudding!Problem 1.3 Robust additionThe text offered addition as an example of intelligent redundancy: We oftenverify an addition by by redoing the sum from bottom