(a) You are given a set of n items of sizes a 1 , . . . , a n , and a bin of siz
ID: 3754959 • Letter: #
Question
(a) You are given a set of n items of sizes a1, . . . , an , and a bin of size B . Your goal is to find a maximum cardinality subset of items that all fit inside the bin. That is, you want to find a set of distinct indices I = {i1, . . . , ik} {1, . . . , n}, such that ai_1 + . . . + ai_k B, maximizing k.
For example, if the sizes are a1 = 7, a2 = 4, a3 = 5, and the bin size is B = 10, then the optimum solution is I = {2, 3} (that is, picking the second and the third item). Design a greedy algorithm for this problem. The running time of your algorithm should be polynomial in n.
(b) Suppose that instead of maximizing k, we want to maximize the total size of the items in the bin; that is, we want to maximize the quantity size(I) = ai_1 + . . . + ai_k . Show that your greedy algorithm does not work in this case.
Explanation / Answer
Trophic status is a useful means of classifying lakes and describing lake processes in terms of the productivity of the system. Basins with infertile soils release relatively little nitrogen and phosphorus leading to less productive lakes, classified as oligotrophic or mesotrophic.
The quantities of nitrogen, phosphorus, and other biologically useful nutrients are the primary determinants of a body of water's trophic state index (TSI). Nutrients such as nitrogen and phosphorus tend to be limiting resources in standing water bodies, so increased concentrations tend to result in increased plant growth, followed by corollary increases in subsequent trophic levels .Consequently, a body of water's trophic index may sometimes be used to make a rough estimate of its biological condition..Although the term "trophic index" is commonly applied to lakes, any surface water body may be indexed.
Trophic classifications
Lakes are commonly classified according to their trophic state, a term that describes how “green” the lake is as measured by the amount of algae biomass in the water. Three trophic state categories are used to describe lakes as they grow progressively greener: oligotrophic, mesotrophic, and eutrophic. Watershed managers typically do not determine trophic state by directly measuring algae biomass, however. Instead, they indirectly assess it by doing the following: (1) Measuring the levels of nutrients and chlorophyll a in the lake (the primary photosynthetic pigment found in plant cells) (2) Measuring lake water clarity using a Secchi disk
Relationships between Trophic Index (TI), chlorophyll (Chl), phosphorus (P, both micrograms per litre), Secchi depth (SD, metres), and Trophic Class
TI
Chl
P
SD
Trophic Class
<30—40
0—2.6
0—12
>8—4
Oligotrophic
40—50
2.6—20
12—24
4—2
Mesotrophic
50—70
20—56
24—96
2—0.5
Eutrophic
70—100+
56—155+
96—384+
0.5—<0.25
Hypereutrophic
Oligotrophic lakes generally host very little or no aquatic vegetation and are relatively clear, while eutrophic lakes tend to host large quantities of organisms, including algal blooms. Each trophic class supports different types of fish and other organisms, as well. If the algal biomass in a lake or other water body reaches too high a concentration (say >80 TI), massive fish die-offs may occur as decomposing biomass deoxygenates the water.
Example current Scenario: Determining Existing Trophic State:
Lake Mesotroph is a pristine Mid-Atlantic Piedmont lake. The lake surface area is 10 square miles, and has an average depth of 20 feet. Its 250 square mile watershed is entirely forested. The county government and the state jointly own much of the land in this watershed. In order to stimulate the local economy, these governments are considering a sale of the property, to be developed as two-acre lots over approximately 90 square miles of the watershed (watershed imperviousness of about 5%). These homes would be seasonal, primarily operating during half of the year, and served by septic systems. A study is being conducted to estimate the impacts of this potential development, and in particular whether the change would shift the lake trophic status. Existing Conditions Monitoring in the area suggests that the forested land use exports only 0.1 lbs/acre of phosphorus per year, and that total streamflow represents approximately 15 watershed inches of runoff per year. The flushing rate, p, of the lake is determined as: p = (15 in/year)(250 mi2)/[(12 in/ft)(10 mi2 )(20 ft)] = 1.55/yr and the flushing rate times lake mean depth, pZ, is determined as: pZ = 1.55/yr x 20 ft = 31ft/yr With the current land use, and including atmospheric deposition, the total annual load to the lake is 19,200 lbs/year. With the lake area of 10 square miles (6,400 acres), the current lake areal load is 3 lbs/acre/ye
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.